Skip to main content

Full text of "Quality index and kinematic analysis of spatial redundant in-parallel manipulators"

See other formats


QUALITY INDEX AND KINEMATIC ANALYSIS OF 
SPATL\L REDUNDANT IN-PARALLEL MANIPULATORS 



i - ' - 
:■' *: '' if:' 



^K 



By 

YU ZHANG 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 

2000 



Copyright 2000 

by 

Yu Zhang 



To my wife, Ying, and our parents. 



ACKNOWLEDGMENTS 

I want to express my deep and sincere gratitude to Dr. Joseph Duffy, my 
supervisor during my Ph.D. study, for providing me with the opportunity to complete my 
study under his exceptional guidance. Without his untiring patience, constant 
encouragement, guidance and knowledge this work would not have been possible. I 
would also like to thank my supervisory committee members. Dr. Carl D. Crane, Dr. 
Gloria J. Wiens, Dr. Ali A. Seirig, and Dr. Ralph Selfridge. I am grateful for their 
willingness to serve on my committee, providing me help whenever needed and for 
reviewing this dissertation. I especially thank Professor Chonggao Liang of Beijing 
University of Posts and Telecommunications for educating me on the various aspects of 
mechanism analysis and design. Also, I would like to thank all my colleagues in the 
Center for Intelligent Machines and Robotics for their help and support. 

Finally, I would like to thank my lovely wife, Ying Zhu. Her love, support and 
encouragement has had made my life rich and complete. I am grateful to my parents and 
parents-in-law for their constant support and encouragement throughout my educational 
endeavors. 



IV 



TABLE OF CONTENTS 

ACKNOWLEDGMENTS iv 

LIST OF TABLES vii 

LIST OF FIGURES viii 

ABSTRACT xi 

L INTRODUCTION 1 

1.1 Redundant Parallel Manipulators 1 

1.2 Quality Index 6 

1.3 Outline of Dissertation 9 

2. SPATL\L GEOMETRY AND STATICS 12 

2.1 Pliicker Line Coordinates 12 

2.2 Statics of a Rigid Body 15 

2.3 The Statics of a Parallel Manipulator 20 

3. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

4-4 IN-PARALLEL MANIPULATOR 24 

3.1 Determination of -ydet J^J^ 25 

3.2 Implementation 30 

4. THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 

4-4 IN-PARALLEL MANIPULATOR 43 

4.1 Inverse Kinematic Analysis 44 

4.2 Forward Kinematic Analysis 47 

4.2.1 Introduction 47 

4.2.2 Coordinate Transformations 48 

4.2.3 Constraint Equations 51 

4.2.4 The Solution 52 

4.2.5 Numerical Verification 55 



5. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

4-8 IN-PARALLEL MANIPULATOR 58 

5.1 Determination of ydet J^J^ 59 

5.2 Implementation 64 

6. THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 

4-8 IN-PARALLEL MANIPULATOR 79 

6.1 Forward Kinematic Analysis 79 

6.2 Numerical Verification 82 

7. THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

8-8 IN-PARALLEL MANIPULATOR 85 

7.1 Determination of ^det J^J^ 85 

7.2 Implementation 94 

8. THE FORWARD KINEMATIC ANALYSIS OF THE SPATL\L REDUNDANT 

8-8 IN-PARALLEL MANIPULATOR 1 10 

8.1 Coordinate Systems 110 

8.2 Constraint Equations 1 13 

8.3 Equation Solution 1 14 

8.4 Numerical Verification 118 

9. CONCLUSIONS 121 

APPENDIX A: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF 
THE REDUNDANT 4-4 IN-PARALLEL MANIPULATOR 123 

APPENDIX B: CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF 
THE REDUNDANT 8-8 IN-PARALLEL MANIPULATOR 130 

LIST OF REFERENCES 148 

BIOGRAPHICAL SKETCH 152 



VI 



LIST OF TABLES 

Table Page 
Table 4.1: Numerical results of the redundant 4-4 in-parallel manipulator 56 

Table 4.2: A numerical example for the special case of the redundant 4-4 in-parallel 

manipulator 57 

Table 6. 1 : Numerical results of the redundant 4-8 in-parallel manipulator 83 

Table 6.2: A numerical example for the special case of the redundant 4-8 in-parallel 

manipulator 84 

Table 8.1: Numerical results of the redundant 8-8 in-parallel manipulator 119 

Table 8.2: A numerical example for the special case of the redundant 8-8 in-parallel 

manipulator 119 



vu 



LIST OF FIGURES 

Figure Page 

Figure 1.1: A planar parallel x-y manipulator with one redundant actuator 2 

Figure 1.2: A 2-DoF planar parallel manipulator 3 

Figure 1.3: A redundant 2-DoF planar parallel manipulator 3 

Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators 4 

Figure 1 .5: Self-deployable space structure 1 1 

Figure 2.1: Determination of a line 13 

Figure 2.2: Pliicker line coordinates 15 

Figure 2.3: Representation of a force on a rigid body 16 

Figure 2.4: Dyname and wrench 19 

Figure 2.5: A 6-6 in-parallel manipulator 21 

Figure 3.1: A redundant 4-4 in-parallel manipulator 24 

Figure 3.2: Plan view of the redundant 4-4 in-parallel manipulator 24 

Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel 

manipulator with the maximum quality index 30 

Figure 3.4: Quality index for platform vertical movement 31 

Figure 3.5: Quality index for platform horizontal translation 34 

Figure 3.6: Platform rotations about the ^''-axis 35 

Figure 3.7: Quality index for platform rotations about the a:'- and >''-axes 37 

Figure 3.8: Platform rotations about the z-axis 38 



Vlll 



Figure 3.9: Quality index for platform rotation about the z-axis 40 

Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel 

manipulator when 6z = 90° 41 

Figure 4.1: Coordinate systems of a redundant 4-4 in-parallel manipulator 45 

Figure 4.2: Coordinate transformations 49 

Figure 5. 1 : A redundant 4-8 in-parallel manipulator 58 

Figure 5.2: Plan view of the redundant 4-8 in-parallel manipulator 58 

Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel 

manipulator with the maximum quality index 63 

Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators. 64 

Figure 5.5: Quality index for platform vertical movement 65 

Figure 5.6: Quality index for platform horizontal translation with different values of j5. 70 

Figure 5.7: Platform rotation about they '-axis 71 

Figure 5.8: Quahty index for platform rotations about the jc'- and j'-axes 73 

Figure 5.9: Platform rotation about the z-axis 74 

Figure 5.10: Quality index for platform rotation about the z-axis 76 

Figure 5.1 1: Plan view of the singularity position of the redundant 4-8 in-parallel 

manipulator when ^^ = 90° 77 

Figure 6.1: Coordinate systems of a redundant 4-8 in-parallel manipulator 80 

Figure 6.2: Leg relations 81 

Figure 7.1: A redundant 8-8 in-parallel manipulator 86 

Figure 7.2: Plan view of the redundant 8-8 in-parallel manipulator 86 

Figure 7.3: Plot of /(a, /3) = 2aj3-2a-2/3 + \ = 90 

Figure 7.4: Plot of /i vs. aand yff with a = 1 92 

Figure 7.5: Plot of V^et J^ J^ vs. av/kha= 1 93 

Figure 7.6: An example of redundant 8-8 manipulator in optimal configuration... 93 



IX 



Figure 7.7: Quality index for platform vertical movement 96 

Figure 7.8: Reduction of the size of the redundant 8-8 in-parallel manipulator 96 

Figure 7.9: Quality index for platform horizontal translation with different values of alOO 

Figure 7.10: Platform rotations about the >' '-axis 102 

Figure 7.1 1: Quality index for platform rotations about the x'- and >' '-axes 105 

Figure 7.12: Platform rotations about the z-axis 106 

Figure 7.13: Quality index for platform rotation about the z-axis 108 

Figure 7.14: Plan view of the singularity position of redundant 8-8 in-parallel 

manipulator when 6^ = 90° 109 

Figure 8.1: Coordinate systems of a redundant 8-8 in-parallel manipulator Ill 



Abstract of Dissertation Presented to the Graduate School 

of the University of Florida in Partial Fulfillment of the 

Requirements for the Degree of Doctor of Philosophy 

QUALITY INDEX AND KINEMATIC ANALYSIS OF 
SPATIAL REDUNDANT IN-PARALLEL MANIPULATORS 

By 

Yu Zhang , ,1 

December 2000 ^ ^■>. t 



Chairman: Dr. Joseph Duffy • ' - 

Major Department: Mechanical Engineering 

Parallel manipulators have been the subject of much investigation over the last 
decade because of their inherent advantages of load carrying capacity and spatial rigidity 
compared to serial manipulators. Usually they have the same number of actuators as their 
degree of freedom, but in some cases, it may be interesting to have more actuators than 
needed and to consider redundant parallel manipulators. Redundancy in actuation can be 
used to increase dexterity, to reduce or even eliminate singularities, to increase reliability, 
to simplify the forward kinematics, and to improve load distribution in actuators. The 
purpose of this work is to design and analyze several spatial redundant parallel 
manipulators. 

The proposed quality index will assist a designer to choose the relative 
dimensions of the fixed and moving platforms, locate joint centers in the fixed and 
moving platforms, determine an optimum position which would be an 'ideal' location of 



XI 



the workspace center, and determine acceptable ranges of pure translations and pure 
rotations for which the platform is stable. 

The quality index for redundant parallel manipulators is defined as a dimensionless 
ratio that takes a maximum value of 1 at a central syimnetrical configuration that is 
shown to correspond to the maximum value of the square root of the determinant of the 
product of the manipulator Jacobian by its transpose. The Jacobian matrix is none other 
than the normalized coordinates of the leg lines. When the manipulator is actuated so that 
the moving platform departs from its central configuration, the determinant always 
diminishes, and, as is well known, it becomes zero when a special configuration is 
reached (the platform then gains one or more uncontrollable freedoms). It is shown that 
the quality index A, for which 0<A< 1, can be used as a constructive measure of not only 
acceptable and optimum design proportions but also an acceptable operating workspace 
(in the static stability sense). 

We also studied the forward kinematic analysis of the redundant in-parallel 
manipulators to determine the position and orientation of the platform, given the leg 
lengths. 



xu 



CHAPTER 1 
INTRODUCTION 

Parallel manipulators have been studied extensively over the last decade with 
their high structural stiffness, position accuracy and good dynamic performance. Usually 
they have the same number of actuators as their degree of freedom, but in some cases, it 
may be interesting to have more actuators than needed to overcome disadvantages of the 
nonredundant parallel manipulators shown by Merlet [25]. 

1 . 1 Redundant Parallel Manipulators 

A number of redundant parallel manipulators have been studied in literature, for 
example, the development of a direct-drive redundant parallel manipulator for haptic 
displays by Buttolo and Hannaford [2, 3], the design of a 2-DoF parallel manipulator 
(Figure 1.1) with actuation redundancy for high speed and stiffness-controlled operation 
by Kock and Schumacher [17], and the addition of a redundant (fourth) branch to three- 
branch manipulators for the purpose of uncertainty elimination and assembly mode 
reduction by Notash and Podhorodeski [28]. 

Maeda et al. [24] also designed a redundant wire-driven parallel manipulator that 
is suitable especially for high speed assembling of lightweight objects such as 
semiconductors. By studying a parallel machining center, O'Brien and Wen [29] 
examined the effectiveness of singularity modification through redundant actuation and 
suggested that augmenting the actuation of a mechanism provides a mechanically feasible 
means of increasing kinematic manipulability. 




Figure 1.1: A planar parallel x-y manipulator with one redundant actuator 

(Adapted from: Kock and Schumacher, A Parallel X-y Manipulator with Actuation 
Redundancy for High Speed and Active Stiffness Applications (1998) [17]) 



Leguay-Durand and Reboulet [23] studied a redundant spherical parallel 
manipulator and showed that actuator redundancy removes singularities and improves 
dexterity in an enlarged workspace. Using a conditioning measure, they compared the 
redundant spherical parallel manipulator with an equivalent nonredundant structure and 
found notably improved uniformity of dexterity for the redundant structure. Similar 
results were also found by Kurtz and Hay ward [19]. 

Kokkinis and Millies [18] found that actuation redundancy allows the selection of 
optimal joint torque for a given load. Nakamura and Ghodoussi [27] also showed that the 
redundant actuation could increase the payload and improve the dynamic response of 
manipulators. 

Dasgupta and Mruthyunjaya [6, 7] saw the redundancy of parallel manipulators as 
the series-parallel dual part of redundancy in serial manipulators. They proposed the 



concept of force (static) redundancy for redundancy in parallel manipulators in contrast to 
kinematic redundancy (widely studied in literature) in serial manipulators. 

In summary, redundant parallel manipulators have the following advantages: 
1. Redundancy in actuation can be used to increase dexterity and reduce or even 
eliminate singularities of parallel manipulators (Pemg and Hsiao [30]). Usually, 
parallel manipulators have a high stiffness, except in some special positions or 
postures where the platform has self-motion and may even collapse. These singular 
configurations may cause serious damage to the manipulator and/or objects in its 
environment. Redundant legs can be used to pull out the platform from singularity 
positions. For example, for the 2-DoF planar parallel manipulator shown in Figure 
1.2, its singularity positions can be found on the line joining the two fixed pivot 
positions, and this singularity can be eliminated by adding another leg as shown in 
Figure 1.3. The new redundant parallel manipulator is entirely free from singularity as 
long as its three fixed pivots are noncollinear. 




|6C=^ 




Figure 1 .2: A 2-DoF planar parallel Figure 1 .3: A redundant 2-DoF planar 

manipulator parallel manipulator 



Here is another example, considering the two cases of spatial nonredundant 4-4 
in-parallel manipulators shown in Figure 1 .4. They are in singularity posifions when 



their platforms are parallel to the base. Such singularity is dangerous because it is not 
immediately obvious from its configuration and if we build such manipulators, they 
may collapse immediately when their platforms are parallel to the base. One possible 
solution is to add another two legs to form a redundant 4-4 in-parallel manipulator 
(Chapters) (Figure 3.1). 





(a) (b) 

Figure 1.4: Planar view of spatial nonredundant 4-4 in-parallel manipulators 



2. Redundancy in actuation can be used to increase the reliability of in-parallel 
manipulators (Shin and Lee [33]). That is, even if some of the actuators fail, a 
manipulator can still operate normally as long as the number of operating actuators is 
not less than the mobility of the manipulator. Thus such a redundant system has a 
failure tolerance, which is increasingly important in robotics, especially when robots 
and manipulators are used in remote or harsh environments such as space, deep sea, 
nuclear plants and for bomb disposal. Because these environments do not allow 



~H~5:/"''<'^ 



immediate human intervention for repair or recovery, the ability of a robot or a 
manipulator to cope with the failures becomes desirable. 

3. The information from the length of the redundant legs can be used to simplify the 
forward kinematics. When controlling a parallel manipulator, we need to do the 
forward kinematic analysis, i.e., to determine the configuration of the moving 
platform given all the leg lengths. This analysis is usually difficult as it involves a set 
of nonlinear equations and, generally, there is more than one solution. For example, 
the forward analysis for the general 6-6 parallel manipulator requires the solution of a 
40"^ degree polynomial (Raghavan [31]) the solution of which is clearly impractical 
for real-time implementation. The additional information from the redundant legs 
reduces many uncertainty positions and even can obtain a unique solution to the 
forward analysis. 

4. The actuator forces and joint torques in the redundant parallel manipulators are not 
uniquely determined. This characteristic can be used to optimize some criteria. For 
example, the joint torque required for a given motion can be minimized. Accordingly, 
it is possible to increase the payload of a closed-link mechanism by adding redundant 
actuators. 

Some other advantages of using redundant actuators are increasing workspace 
while improving dexterity, having autonomous calibration, and building variable 
geometry trusses. Possibilities of redundancy in parallel manipulators and their effective 
use have not been studied extensively until now. The purpose of this work is to design 
and analyze several spatial redundant parallel manipulators. 



1.2 Quality Index 
Parallel manipulators have better load carrying capacity and spatial rigidity than 
serial manipulators. However, the complexity of the kinematics of parallel manipulators 
makes it more difficult for a designer to determine a set of kinematic and geometry 
parameters that will efficiently produce prescribed performances. Indeed, the behavior of 
parallel manipulators is far less intuitive than that of serial manipulators. The geometric 
properties associated with singularities, for example, may be much more difficult to 
identify directly (Fichter [10] and Merlet [26]). Therefore, more systematic analysis and 
optimization tools are needed to make parallel manipulators more accessible to designers. 
At this time little information is available to assist designers in the following task: 

(a) Choose the relative sizes of the fixed and moving platforms. 

(b) Locate the positions of the centers of the spherical joints in the base and the 
centers in the moving platform. 

(c) Determine an optimum position that would be an ideal 'center' location of the 
workspace. 

(d) Determine acceptable ranges of pure translations of the platform for which the 
platform is stable (i.e., not too close to a singularity). However, the question 
"How close is too close?" is often hard to answer. 

(e) Determine acceptable ranges of pure rotations of the platform for which the 
platform is stable. 

(f) Determine the ranges of leg displacements. 

These considerations are the reasons that the quality index was proposed. 

The quality index was defined initially for a planar 3-3 in-parallel device by the 
dimensionless ratio (Lee, Duffy, and Keler [22]) 



> }'.- ; ■.■ 



, detj 

>^ = i' 'i (1.1) 

where J is the three-by-three Jacobian matrix of the normalized coordinates of three leg 
lines. Then it was defined for an octahedral in-parallel manipulator by Lee, Duffy, and 
Hunt [21] and 3-6, 6-6 in-parallel devices by Lee and Duffy [20]. For these cases J is the 
six-by-six matrix of the normalized coordinates of the six leg lines. For these fully 
synmietrical nonredundant parallel manipulators the quality index takes a maximum 
value of /I = 1 at a central symmetrical configuration that corresponds to the maximum 
value of the determinant of the six-by-six Jacobian matrix (i.e., det J = det 7m) of the 
manipulator. When the manipulator is actuated so that the moving platform departs from 
its central configuration, the determinant always diminishes, and, as is well known, it 
becomes zero when a special configuration is reached (the platform then gains one or 
more uncontrollable freedoms). 

In this dissertation, the quality index is extended for redundant manipulators by 
the dimensionless ratio 



detJJ^ 
VdetJ^J, 



' m" m 



This makes complete sense because the Cauchy-Binet theorem det JJ^ =A^ +A2 -I---+A^ , 



has geometrical meaning. Here, each A, ( 1 < / < m = 






) is simply the determinant of the 



6x6 submatrices of J which is a 6xn matrix. This is clear when w = 6, (1.2) reduces to 
(1.1). It has been shown by Lee et al. [21] that by using the Grassmann-Cayley algebra 
(White and Whiteley [35]), for a general octahedron, when the leg lengths are not 
normalized, det J has dimension of (volume)^ and it is directly related to the products of 



8 

volumes of tetrahedra that form the octahedron. In this way detj and vdet JJ^ have 
geometrical meaning. 

We mention in passing the work of Cox [4] and Duffy [8], both of which cover 
special configurations of planar motion platforms. Hunt and McAree [14] go into 
considerable detail regarding the general octahedral manipulator. Its special 
configurations are described in the context of other geometrical properties. A few papers 
were published on the optimal design of nonredundant parallel manipulations (see for 
example Gosselin and Angeles [11, 12], Zanganeh and Angeles [36]). 

Zanganeh and Angeles [36] point out problems with quantities such as condition 
number due to the inherent inhomogeneity of the columns of the Jacobian, /. This is 
precisely why equations (1.1) and (1.2) are adopted as an index of quality rather than 
other well-established methods (found in books on theory of matrices and linear algebra) 
that lead (via norms, diagonalization and singular values, etc.) to properties that relate to 
'conditioning'. All such methods are based implicitly on the presumption that a column- 
vector (say, of a six-by-six matrix) can be treated as a vector in 9?^. However, the six 
elements in the column of a typical robot Jacobian are the normalized coordinates of a 
screw (almost always of zero pitch; i.e., a line); in a metrical coordinate frame three of 
them are dimensionless and three have dimension [length], such a length being the 
measure of the moment about a reference point of a unit force. The column generally 
comprises two distinct vectors (each of them in 91^). For the legs of the nonredundant and 
redundant manipulators it is not possible to remove all the length dimensions from their 
coordinates. Even the adoption of some artificial length unit fails, simply because a 
moment can never be converted to a pure force. Moreover, any index of quality derived 



from such textbook techniques is likely to vary according to the coordinate frame in 
which the Jacobian is formulated. Our method works for two reasons: first, the 
determinant of a (square) Jacobian of line coordinates depends solely on the 
configuration in 9t^ of the actuated axes and not on the coordinate frame in which the line 
coordinates are determined. The second reason is that equations (1.1) and (1.2) are 
dimensionless ratios, and our quality indices are always independent of the choice of 
units of length measurement. 

Unlike the case of a mechanism designed for a specific task, the tasks to be 
performed by a manipulator are varied. Hence, there should not be any preferred general 
orientation for which the manipulator would have better properties. It suggests that the 
manipulator should be symmetrical. Such symmetrical configurations may not always 
exist, of course. However, except for unusual applications (and there will undoubtedly be 
some where for example unusual loads must be sustained) we are safe in seeking 
centrally symmetrical designs to which we can assign the highest quality index /l=l, or 
close to it. For these cases, contours of quality index help to determine a realistic 
workspace volume that is free from singularities. Therefore, we are concerned primarily 
with symmetrical redundant parallel manipulators in this dissertation. 

1 .3 Outline of Dissertation 

A simple introduction to the screw theory is presented in Chapter 2 to provide 
insight into how a screw-based Jacobian matrix of a parallel manipulator is determined. 

In Chapters 3 and 4 a spatial redundant 4-4 in-parallel manipulator is studied first. 
The device consists of a square platform and a square base connected by eight actuated 
legs. As in Chapter 3, the quality index for the redundant 4-4 parallel manipulator is 



10 

determined. To achieve the maximum quality index for a redundant 4-4 in-parallel 
manipulator with platform side a, the base has side 42a and the perpendicular distance 

between the platform and the base is —j= . The kinematic analysis of the redundant 4-4 

V* 

in-parallel manipulator is studied in Chapter 4. The derivation of forward kinematic 

equations for position and orientation of the platform is described. 

Chapters 5 and 6 extend the study to a redundant 4-8 in-parallel manipulator with 
a square platform and an octagonal base. The octagonal base is formed by separating 
from each vertex of a square by a small distance. The quality index for this manipulator is 
determined in Chapter 5. The compatibility between the redundant 4-4 and the 4-8 
parallel manipulators also is discussed in this Chapter. Chapter 6 solves the forward 
kinematics of the redundant 4-8 parallel manipulator by transferring the problem to the 
corresponding redundant 4-4 case. . 

Finally, in Chapters 7 and 8, a redundant 8-8 in-parallel manipulator is studied. 
The device has an octagonal platform and a similar octagonal base connected by eight 
legs. Such arrangement avoids using double-spherical joints because they can produce 
serious mechanical interference. However, by using the quality index determined in 
Chapter 7, the best design can be obtained when the pair of separated joints in the base 
and top platform are as close as possible. In Chapter 8, the kinematic analysis of the 
redundant 8-8 parallel manipulator is performed. The forward analysis gives a much 
simpler solution than that of the nonredundant case. 

Using quality index, variable motions are investigated for which a moving 
platform rotates about a central axis or moves parallel to the base. The quality index can 
be used as a constructive measure not only of acceptable and optimum design proportions 



11 

but also of an acceptable operating workspace (in the static stability sense). Moreover, 
analysis of these redundant in-parallel manipulators can be used to model and design a 
self-deployable space structure that has a pair of flexible antenna platforms in the base 
and top platform as shown in Figure 1.5 (Duffy et al. [9] and Knight et al. [16]). 





Figure 1.5: Self-deployable space structure 



CHAPTER 2 
SPATIAL GEOMETRY AND STATICS 

Chapter 1 showed that the quaUty index of parallel manipulators is based on the 
Jacobian matrix. This chapter, which is mostly a general background in screw theory 
(Ball [1]), provides insight into how the Jacobian matrix of parallel manipulators is 
determined. Firstly, we review some basic concepts of spatial geometry and screw theory. 

2. 1 Pliicker Line Coordinates 
Two distinct points ri(A:i, y\, z\) and r2(x2, yi, zi) can be connected by a line in 
space. The vector S whose direction is along the line can be written in the form 

S = r2-ri. (2.1) 

Alternatively this may be expressed as 

S = Li + Mj+Nk (2.2) 

where 

L = X2-xi, M = y2-y\, N = Z2-Z\ (2.3) 

are defined as the direction ratios of the line and they are related to the distance ISI 
between the two points by 

L^ + M^ + N^ = \S\^ (2.4) 

where the notation 1 1 denotes absolute magnitude. 
Often L, M, and A^ are expressed in the form 

L= —^ -, M = ^^ — ^, N=— -, (2')) 

ISI ISI ISI ^"^-^^ 

which consists of unit direction ratios of the line, and (2.4) reduces to 

12 



13 



l^ + m'^ + n^=\. 



(2.6) 



If r represents a vector from the origin to any general point on the Une (Figure 2.1), then 
the vector r-ri is parallel to S and therefore the equation of the line can be written as 

(r-ri)xS = (2.7) 



and in the form 



where 



r X S = So 



So = ri X S 



(2.8) 



(2.9) 



is the moment of the line about origin O and is clearly origin dependent. Further, because 
So=rixS, the vectors S and S© are perpendicular and as such satisfy the orthogonality 
condition 



S • So = 0. 



(2.10) 




Figure 2. 1 : Determination of a line 



14 

The coordinates of a line are written as [S; So]' and are referred to as the Plucker 
coordinates of the Hne [13]. The coordinates [S; So] are homogeneous since from (2.8) 
the coordinates [kS; kSo] (k is a non-zero scalar) determine the same line. 

Expanding (2.9) yields 



• 

I 


• 

J 


k 


X\ 


y^ 


Z\ 


L 


M 


N 



So = 



which can be expressed in the form 

So = Pi + OJ + ^k 
where 



(2.11) 



(2.12) 



P= y,N - z,M, 
Q= z^L - x^N, 
R= XiM - y,L. 



;,-, ,^^'t«> 



(2.13) 



From (2.2) and (2.12) the orthogonality condition S-So=0 can be expressed in the form 

LP + MQ + NR = 0. (2.14) 

The Plucker coordinates of the line [S; So] now can be written in terms of their 
components as [L, M, N; P, Q, R], which are known as the ray coordinates for a line 
(Figure 2.2). Unitized coordinates for a line can be obtained by imposing the constraint 
that ISI=1. The Plucker coordinates thus must satisfy equations (2.6) and (2.14) and hence 
only four of the six scalars L, M, N, P, Q, and R are independent. It follows that there are 
°°'^ lines in space^. 



The semi-colon is introduced to signify that the dimension of ISI is different from ISqI. 
Systems of lines and their properties, oo' (hne series), oo^ (congruence), oo^ (complex), 
are described by Hunt [13] which contains an extensive bibliography on the subject. 



13 



-■ V. 




Figure 2.2: Pliicker line coordinates 



A straightforward method to obtain the Pliicker coordinates was given by 
Grassmann (Hunt [13]) by expressing the coordinates of the points ri(xi, _yi, zi) and rjfe, 
J2, Zi) in the array 



1 X, y^ z, 
1 Xj y^ Zj 

and by expanding the sequence of 2x2 determinants 



L = 



1 X, 


» 


M = 


1 y. 


iV = 


1 ^. 


f 


1 x^ 




1 ^2 




1 z. 




y^ z, 
yi ^2 




, Q = 


Z, JC, 

Z2 ^2 


/? = 


^2 y: 





(2.15) 



(2.16) 



2.2 Statics of a Rigid Body 
The concepts developed in the previous section now can be applied directly to the 
statics of a rigid body. A line $ with ray coordinates [S; So] (where ISI = 1) can be used to 



16 

express the action of a force upon a body (Figure 2.3). Because the body is rigid, the 
point of application can be moved anywhere along the line. 




Figure 2.3: Representation of a force on a rigid body 



As illustrated by Figure 2.3, a force f can be expressed as a scalar multiple^ of 
the unit vector S that is bound to the line $. The moment of the force f about a reference 
point O is itio which can be written as mo = rxf where r is a vector to any point on the 
line $. This moment can also be expressed as a scalar multiple /So where So is the 
moment vector of the line $ (i.e.. So = rxS). The action of the force upon the body thus 
can be expressed elegantly as a scalar multiple /$ of the unit line vector, and the 
coordinates for the force are given by 

/$=/[S;So] = [f;ino] (2.17) 

where SS=1 and SSo=0. 



17 

Clearly, when the reference point O is coincident with A, then ino=0 and the 
coordinates of the force are [f; 0]. Therefore, f is a line bound vector that is invariant with 
a change of coordinate systems while mo is origin dependent. 

An important special case is [0; mo] which can be considered as the resultant of a 
pair of equal and opposite forces with coordinates [f; moi] =/[S; Soi] and [-f; moa] =/[-S; 
S02], where ISI=1. The coordinates of the resultant [0; ma] = [0; moi+moa] =/[0; S01+S02] 
are not a line bound vector, but a pure couple. The couple can be considered as equivalent 
to a force ^ of infinitesimal magnitude (l^-^O) acting along a line that is parallel to the 
lines of action of the pair of parallel forces. The line of action of 8i is infinitely distant 
with coordinates [0; mo], such that lpl = oo where p is the vector from the origin 
perpendicular to the line of action of 51, and the moment of the force <5f about the origin 
is px^=mo. A pure couple thus can be represented as a scalar multiple of a line at 
infinity. 

The problem of determining the resultant of an arbitrary system of forces with 
coordinates [fi; moi], [fj; moa], ..., [fn; mon] acting on a rigid body is essentially the 
determination of the quantity 

w=[f;mo], (2.18) 

where 

n n 

f=5^fi and m„=^m„,. (2.19) 

1=1 .=1 

It is assumed at the outset that a reference point O was chosen so that the forces 
acting on the rigid body were translated to point O and so that moments moi, mo2, ..., mon 
were introduced to yield an equivalent system of forces and torques that act on the rigid 



18 

body. Therefore, the line of action of the resultant force f passes through point O and the 
resultant moment m© is a couple [0; mo]. In general f and mo are not perpendicular (i.e., 
f-mo^K)). The new quantity with coordinates w = [f; mo] therefore is not a force and was 
defined as a dyname by Plucker. 

Because in general f • m© ;t 0, it is not possible to translate the line of action of 
force f through some point other than point O and to have the translated force produce the 
same net effect on the rigid body as the original dyname. The moment mo, however, can 
be resolved into two components, ma and mt, which are respectively parallel and 
perpendicular to f (Figure 2.4a) and 

mo = ma + mt . (2.20) 

The moment m, can be determined as 

ma = (mo -8)8 (2.21) 

where 8 is a unit vector in the direction of the resultant force f. The moment mt is then 
determined as 

m, = mo - ma. *^ (2.22) 

The line of action of force f now can be translated so that the force with coordinates 
[f;mt] plus the moment [0;ma] (Figure 2.4b) is equivalent to the dyname [f;mo]. 
Therefore, the dyname is represented uniquely by a force f acting on the line [8; 8ot] 

(where 8ot=-7^) and a parallel couple ma. This parallel force-couple combination was 

called a wrench by Ball [1]. 

From (2.18) and (2.20), the wrench w which is equivalent to the dyname [f; mo] 
can be expressed in the form 



19 

w = [f ; mo] = [f ; mt + ma] = [f ; mt] + [0 ; ma]. 
Clearly, [f; mt] is a pure force because f •mt=0. 



(2.23) 




(a) Dyname, [f; mo] (b) Wrench, [f; mt] + [0; ma] 

Figure 2.4: Dyname and wrench 



Further, because ma is parallel to f, then • ' ' ' 

ma = /if (2.24) 

where his a non-zero scalar which is called the pitch of the wrench. From (2.20) 

f • mo = f • ma (2.25) 

and from (2.24) and (2.25), the pitch h is given by 

f m„ f m„ 



h = 



^ _ * '"0 

f f f f 



(2.26) 



Substituting (2.24) into (2.23), together with (2.22) allows the wrench w to be 
written as 



w=[f ;mo-/?fl + [0;/zfl. 



(2.27) 



.20 ,; _:-. . ,..,^^^, 

Thus, the coordinates for the Hne of action of the wrench are [f; itio-Zif] and from (2.27) 
the equation for the hne is 

rxf=mo-/if. (2.28) 

In the same way as the action of a force can be expressed as a scalar multiple of a 
unit line vector, a wrench can be expressed elegantly as a scalar multiple of a unit screw $ 
where 

$ = [S;So] (2.29) 

and where SS= 1 . From (2.26), the pitch of the screw is given by 

h = SSo. (2.30) 

Further, from (2.2) and (2. 1 2), 

h = LP + MQ + NR. (2.31) 

Therefore, Ball [1] defined a screw as "a line with an associated pitch". Following (2.28), 
the Plucker coordinates for the screw axis are [S; Sq-ZiS] and the equation of the axis is 

r X S = So — ^S. ^2 32) 

2.3 The Statics of a Parallel Manipulator 
Figure 2.5 illustrates a nonredundant 6-6^ in-parallel manipulator. The device has 
a moving platform and a fixed base connecting by six legs each of which is the same 
kinematic chain. The prismatic joint in each leg is actuated and the moving platform has 
six degrees of freedom. 



■a 

These numbers indicate the number of connecting points in the top and base platform 
respectively. 



21 




moving 
platform 



base 



Figure 2.5: A 6-6 in-parallel manipulator 



Consider that the six leg forces with magnitudes /i,/2, ...,/6 are generated in each 
of the lines $i, $2, .... $6- The resultant wrench w = [f; mo] acting upon the moving 
platform due to these six leg forces is given by 

w = [fi; moi] + [f2; nioi] + . . . + [fe; mo6], (2.33) 

or in the alternative form, 



w=/i[Si; Soi] +/2[S2; S02] + ... +/6[S6; Soe] 



(2.34) 



where [Si; Soi] (ISil=l, /=1...6) are the Pliicker line coordinates of the six legs. Further, 
(2.34) can be expressed in the matrix form 

w = JF (2.35) 

where w = [f , hIq ]^ and F = [/, , /a , /a , A , /s , f^V are 6x1 column vector. J is a 6x6 
matrix of line coordinates given as 



J = 



22 



S, Sj S3 S4 Sj Sj 

^01 ^02 ^03 ^04 ^05 ^06 



(2.36) 



and is called the Jacobian matrix, or simply Jacobian, which enables us to determine the 
resultant wrench w= [f; m©] produced by six actuator forces generated in the legs. It 
should be noted that for redundant parallel manipulators the Jacobian matrix is not 
square. 

The transpose of the Jacobian matrix relates the infinitesimal displacements 5l\ in 
each leg to the infinitesimal displacement twist"* of the platform and 

6\ = i^6t) (2.37) 

where S\ = [6l^,6l2,...,5lf^f and dh = [5x,dy,5z,5(p^,6(Py,S(p^f . Here, dx, Sy, 

and Sz are the infinitesimal displacement of a point in the moving platform coincident 
with a reference point O which is chosen to be the origin of a fixed coordinate system on 
the fixed base. The quantities S<p^ , d(p^ , and Scp^ are infinitesimal rotations of the 

moving platform about the axes of the fixed reference coordinate system. 

In summary, the Jacobian matrix of parallel manipulators serves two distinct 
purposes. In its ordinary form the columns of which are the coordinates of the actuator 
lines (normalized), it enables us to obtain from actuated force inputs the wrench at the 
end effector platform. In its transposed form the Jacobian can give the relative speeds 
required at each actuator that corresponds to a given twist to be executed by the platform. 
The first of these gives the instantaneous solution to a problem of static equilibrium; the 
second, the solution of first order kinematic compatibility. When the Jacobian matrix is 



An infinitesimal twist is also a scalar multiple of a unit screw, as the scalar is an 
infinitesimal rotation with unit of radian. 



23 

singular (i.e., its rank is less than six) the actuators (i) cannot equilibrate a general wrench 
applied to the platform and (ii) cannot on their own prevent a transitory uncontrollable 
movement of the platform. This latter phenomenon is associated with the platform's 
gaining one or more freedoms when all the actuators are locked. The platform is then in a 
singularity position. 



CHAPTER 3 

THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

4-4 IN-PARALLEL MANIPULATOR 

A spatial redundant 4-4 in-parallel manipulator is shown in Figure 3.1. The device 

has a square platform and a square base connected by eight legs. Figure 3.2 shows a plan 

view of this manipulator, where the moving platform is symbolically represented by four 

pairs of concentric spherical joints located at points A, B, C, and D, and the fixed base is 

represented by another four pairs of concentric spherical joints located at points E, F, G, 

and H. This manipulator is said to be redundant since the platform and the base are 

connected by eight actuated legs. 





Figure 3.1: A redundant 4-4 in-parallel 
manipulator 



Figure 3.2: Plan view of the redundant 4-4 
in-parallel manipulator 



24 



25 



3.1 Determination of ^det.T^.T^ 



The moving platform of the redundant 4-4 parallel manipulator shown in Figure 
3.2 is located at its central symmetrical configuration and is parallel to the base with a 
distance h. At this configuration, the manipulator is fully symmetric and each leg has the 
same length. Clearly, at such position the platform is most stable from the geometric 
static point of view. When the platform departs from this central symmetric position, the 
platform will lose its geometric symmetry and the eight leg lengths will be different. 
Therefore, it is reasonable to assume that at the central symmetric configuration shown in 
Figure 3.2, it is possible to determine the values of square base side b and height h based 
on square platform side a so that a maximum value of the square root of the determinant 

of the product of the manipulator Jacobian by its transpose, i.e., ^det J^J^ , may be 

obtained. 

Firstly, the coordinates of the points A, B, C, and D on the platform and E, F, G, 
and H on the base are determined with the origin of a fixed coordinate system placed at 
the center of the square base as shown in Figure 3.2, and 



- 



V2fl 



, B 



\ 



- — 

2 2 



4ia 



(b 



\ ( 



h 



, C 







V2a 



\ 



-- 

2 2 



2 2 



, D 



H 



yfla 



h 



(3.1) 



/- 



-* t 

2 2 



A 



Then, using the Grassmann method described in Chapter 2 to calculate the 
Pliicker line coordinates of the eight leg lines, i.e., counting the 2x2 determinants of the 
various arrays of the joins of the pairs of points EA, FA, FB, GB, GC, HC, HD, and ED. 



26 

For example, the coordinates of the Hne $i are obtained using the coordinates of points E 
and A in (2. 15) to form the array 



1 -^ 

2 

1 



2 
4la 







(3.2) 



and using (2.16) to yield 



S,= 



'42a -b , bh bh slab 
1 . h\ — — , --, — : — 



(3.3) 



Similarly from points F and A, 

b yfla-b 



S,= 



bh _bh -Jlab 

2 2 2 4 



(3.4) 



From points F and B, 



S, 



y[2a-b b , bh bh ^lab 
— I — ' T' «; --:r. — —^ — : — 



(3.5) 



From points G and B, 



5,= 



42a -b 



2' ' 2' 2' 4 



(3.6) 



From points G and C, 



_^ -jla-b bh _bh yf2ab 

2' 2 ' ' 2' 2' ^~ 



(3.7) 



From points H and C, 



^6 = 



^ V2a-j? K ^ bh Sab 
2' 2 ' ' 2' 2' 4~ 



(3.8) 



From points H and D, 



27 



S,^ 



Sa-b _b_ , bh bh ^Jlab 
2 2 2 2 4 



(3.9) 



From points E and D, 



5„ = 



yfla-b 



b 

1 



bh bh 

2' 2 



h; — — , — -, - 



42ab 



(3.10) 



It should be noted that the above Pliicker line coordinates are not normalized and each 
line must be divided by /, = |Si| (/ = 1, 2, ..., 8). Hence, the normalized Jacobian matrix 
of the eight leg lines (now all reduced to unit length) can be expressed as 



J = 



si 


si 


Si 


si 


si 


si 


si 


si 


L^> 


h 


h 


h 


h 


h 


I, 


h 



(3.11) 



Since the device is in a synunetrical position, the normalization divisor is the 
same for each leg, namely /, = /(/= 1,2, . . ., 8), and for every leg 



l = ^L^+M^+N' =J-(a'-yf2ab + b' + 2h'). 



(3.12) 



From (3.3) to (3.10), the Jacobian matrix in (3.11) becomes 



J=- 

/ 



b 
2 

h 

bh 
2 
bh 
2 
y[2ab 



_b 
2 

-d, 



b 
2 



bh _bh 

2 2 

_bh _bh 

2 2 

■\j2ab 42ab 



2 



b_ 

2 



d, 
h 



bh bh 

2 2 

_bh _bh 

2 2 

s2ab 42ab 



2 



-d, 

2 



bh bh 

2 2 

bh bh 

2 2 

\2ab \2ab 



-d, 

b 

2 

h 

bh 

2 
bh 

2 
^ab 



(3.13) 



28 



where 



d,= 



42a-b 



Using equation (3.13), the determinant of the product J J ^ can be expressed in the 



form 



detJJ^=- 



/ 



12 



^2 











-d. 








d. 





d. 














8/1^ 














d. 





Ib'h' 








-d. 











2b'h' 




















aW 



where 

d^=2{a'-42ab+b''), 
d^^{42a-2b)bh. 

Expanding (3.14) and using (3.12), then extracting the square root yields 

(a^-^l2ab + b^+2h^} 



(3.14) 



(3.15) 



Assuming the top platform size a is given, now taking the partial derivative of (3.15) with 
respect to h and b respectively and equating to zero yield 



and 



9642a'b'h'(a'-yf2ab + b^-2h') 
(a^-yf2ab + b^+2h^} 



96Sa'b'h'{a'-b'+2h') _^ 
[a^ -42ab + b'' +2h^)' 



(3.16) 



(3.17) 



29 

When a, b, and h are not equal to zero, equations (3. 16) and (3.17) give 

a^-^Jiab + b^-lh' =Q, (3.18) 

a^-b^+2h'=0. (3.19) 

Adding (3.18) and (3.19) yields 

2a^-42ab = 0. (3.20) 

Solving the above equation, we obtain 

b = 42a. (3.21) 

Further, substituting (3.21) into equation (3.19) yields 

a^-2h^=Q. (3.22) 

There are two solutions for h in the above equation, here we only take the positive 
solution (the negative solution is simply a reflection through the base) 

" = ;!• ... . ;.., . (3.23) 

Finally, substituting (3.21) and (3.23) into (3.15) we get 

VdetJ^J: = {VdetJJ^)^ = A^la' (3.24) 

where Jm denotes the Jacobian matrix for the configuration at which the 4-4 redundant 
parallel manipulator has a maximum quality index. This optimum configuration is shown 
in Figure 3.3. 



30 




Figure 3.3: Plan view of the optimal configuration of the redundant 4-4 in-parallel 
manipulator with the maximum quality index 



3.2 Implementation 
From the definition of quality index (see (1.2)) and (3.24), the quality index for 
the redundant 4-4 parallel manipulator shown in Figure 3.1 becomes 

VdetjJT 



Z = - 



4yf2a^ 



(3.25) 



The variation of the quality index now is investigated for a number of simple motions of 
the top platform. Here, an optimal redundant 4-4 parallel manipulator with platform side 

a = 1 , and thus base side Z> = V2 is taken as an example. 

First, consider a pure vertical translation of the platform from the central 
symmetric position shown in Figure 3.2 along the z-axis while remaining parallel to the 
base. For such movement, from (3.15) and (3.25), the quality index is given by 



Sb'h 



31 



31.3 



{a^-yl2ab + b^+2h^J 
With a = 1 and ^ = v2 , this reduces to 



(3.26) 



A = 



\e42h^ 



(3.27) 



and is plotted in Figure 3.4 as a function of h. It shows that at height h = -p = , the 

-v2 2 

quality index of the redundant 4-4 parallel manipulator has a maximum value, A = 1 . 



1.0 - 
0.8 - 

■S 0.6 - 

c 

3 0.4- 

o 

0.2- 

0.0- 












f\ 










\ 








\ 


1 








\t^ 








i ■ ■ t ■■ 



0.0 0.5 1.0 1.5 2.0 2.5 3.0 
Height h 

Figure 3.4: Quality index for platform vertical movement 



We now derive an expression for -^det JJ^ when the platform of the redundant 
4-4 parallel manipulator is translated away from its central location while remaining 
parallel to the base at height h. Assume the center of the moving platform to move to 
point (x, y, h), then the coordinates of the points A, B, C, and D on the platform become 



32 



X y- 



42a 



X y-\ h 



\ 



B 



D 



x + - 



42a 



X-- 



42a 



y h 



y h 



(3.28) 



The coordinates of points E, F, G, and H on the base can be found from (3.1). Thus, the 
Pliicker line coordinates for each of the eight leg lines can be determined as 

(3.29) 



(3.30) 



s^ = 


b 42a-b 
;"2' '- 2 ' 


h; 


bh 

2' 


bh 

2' 


b{2x-2y + 42a) 
4 


s,= 


b 42a-b 
['-2' '- 2 ' 


h; 


bh 
2' 


bh 
2 


b{2x + 2y-42a) 
4 




42a-b b 


h; 


bh 
2' 


bh 
2 


b{2x + 2y + 42a) 
4 




42a-b b 
2-^2 


h; 


bh 
2' 


bh 
2' 


b{2x-2y + 42a) 
4 


Ss = 


b 42a-b 

^2' '^ 2 ' 


h; 


bh 
2' 


bh 
2' 


b{2x-2y-42a) 
4 




b 42a-b 
^2' '^ 2 ' 


h; 


bh 
2' 


bh 
2' 


b{2x + 2y + 42a) 
4 J' 




42a-b b 

r 2 ' '-2' 


h; 


bh 
2' 


bh 
2' 


b{2x + 2y-42a) 
4 J' 


Ss- 


42a-b b 

r 2 ' '^2' 


h; 


bh 

2' 


bh 

2" 


b{2x-2y-42a) 
4 



(3.31) 



(3.32) 



(3.33) 



(3.34) 



(3.35) 



(3.36) 



33 



The above coordinates are not normalized and each row must be divided by the 
corresponding leg length. The Jacobian matrix J then can be constructed by using (3.11). 

Further, substituting and expanding -^det JJ^ yields 






(3.37) 



where the leg lengths are 




h=- 



li=- 



( ^1 


2 / 


X 


+ 


I 2j 


V 



y+- 



-^a-b 



+ h' 



42a-b"\ ( M' 



X — - 



r-h 



+h' 



l2=- 



h=- 



h = 



/«=. 



( b^ 

X — 
V 2, 



+ 



Sa-b 



+ h\ 



42a-b^ ( b^' 



JC + - 



+ 



r-^, 



+h' 



( ^Y ( 4la-b 



\2 



X + 

V 2, 



y + - 



+ h\ 



^ 4ia-b^ ( b^' 



X-- 



J* 2, 



+ h' 



^ 



With a = \,b = yf2,mdh = , from (3.25) and (3.37), the quality index A becomes 



VdetJJ^ 

^d^iJJl ~ ixU2x'y'+y' +2x'+l){xU2x'y'+y* +2y'+\) ' 



x^+y^+] 



(3.38) 



which is plotted in Figure 3.5(a) as a function of x and y. Figure 3.5(b) shows the 
contours of the quality index for this platform horizontal movement. The contours are 
labeled with values of constant quality index and they are close to being concentric 
circles of various radii. When j: or _y is infinite, A=0, and when x=y=0,X=\. 



34 




1 -1 




-0.5 - 



-1.0 



(a) (b) 

Figure 3.5: Quality index for platform horizontal translation 



To illustrate the variation of the quality index during some simple rotations of the 
platform, a new coordinate system x'y'z' is attached to the square platform. The origin of 
the new coordinate system is located at the center of the top platform and the coordinate 
system is oriented such that its a; '-axis is passing through vertex B, its ^''-axis then is 
passing vertex C, and its z'-axis is normal to the square platform ABCD. Thus, when the 
platform locates at its initial central position shown in Figure 3.2, the x'- and ^''-axes are 
parallel to the x- and >'-axes on the base respectively. 

Figure 3.6 illustrates a side view of the redundant 4-4 parallel manipulator when 
the square platform ABCD is rotated by an angle dy about the y'-axis from its initial 
position. For such platform rotation, the coordinates of the vertices A, B, C, and D 
become 



35 











Sa 



yfla 



\ 


f 


h 


B 


J 


\ 


\ 


( 


» 


D 


/ 


v 



cos^„ h + sin^„ 

2 ^ 2 ^ 



^[2a 



42a 



cos9^. h- 

2 ' 2 



sin 0. 



(3.39) 




Figure 3.6: Platform rotations about the >> '-axis 



It should be noted that the positions of line $i, $2, $5, and $6 do not change during 
this movement and their corresponding Plucker line coordinates can be obtained from 
(3.3), (3.4), (3.7), and (3.8) respectively. The Plucker coordinates for the line $3, $4, $7, 
and $8 are now given by 

^f2a cos 0^. -b b 42a sin 0^. + 2h 



S,= 



2 ' 2' 2 ' 

b(42a sin 0^ + 2h) b{42a sin 0^ + 2h) 42ab cos 0^. 



(3.40) 



36 



^4 = 



■\J2acosdy-b b yflasinO^ +2h 
2~ ' "2' 2 ' 



bi-yfla sin 6^ + 2h) b{42a sin d^ + 2h) y/lab cos 9^ 



4 



(3.41) 



S, 



yfla cos ^^ - ft ft 42a sin ^^ - 2/i 
2~ ' "2' 2" ' 

b(^I2a sin ^^. - 2/i) b{^|2a sin ^ - 2/z) ^f2ab cos 6' 



(3.42) 



5« = 



^J2a cos ^, - ft ^ V2a sin ^^ - 2/i 
2 2 2 



Z7(V2a sin ^^, - 2h) b{42a sin ^^, - 2h) -Jiab cos ^^ 



(3.43) 



Since the configuration of the manipulator keeps symmetric about the x-axis 
during the platform rotation about the j'-axis, from Figure 3.6, we have 
/, = /2 = /j = /g = / , /j = /4 , and /, = /g . The corresponding Jacobian matrix then can be 



determined by (3.1 1), and further ^det JJ^ becomes 

4hi-y/2a-b){l^-l^)sme^+2i4h^cos^e^+b^sm^e^)l^) 

{h\4h^cos'e^ +b^sm^e^)(l^ +l^)-Abh\ll -l^)cose^.^md^ + (3.44) 

((4a'/z' cos' e^ +a^b^ sin' ^^ + 2b^h^ -^^abh" cos^Jsin' ^^ + 

8/i'cos^ )/'))2 



where 



37 



I, =-^{j2acose^ -bf +b^ +{y[2asme^ +2h)\ 



Ij = -^(V2fl cos d^-bf+b^+ (Jla sin 6^ -2h)\ 



and / can be found from (3.12). 



S 



With a = \, b = ^f2, and h = — , from (3.25) and (3.44), the quaUty index 



becomes 



A = 



-y/detJJ^ _ -^(3 - cos 6' J(6 cos ' 0^. - cos ' 6^. - 7 cos ^,, + 4) 
VdetJ™J; ~ 



2(2cos'^^, -4cos^,-l-3) 



(3.45) 



Since the redundant 4-4 in-parallel manipulator is fully symmetric at its central 
configuration shown in Figure 3.2, the same result can be obtained when the platform is 
rotated about the jc'-axis. 

From (3.45), the variation of the quality index for rotations about the x'- and y'- 
axes is drawn in Figure 3.7. 




Rotation Angle (degree) 



Figure 3.7: Quality index for platform rotations about the x'- and y'-axes 



38 



Figure 3.8 illustrates a plan view of the redundant 4-4 parallel manipulator with 
the moving platform ABCD rotated ^ about the z-axis. The x and y coordinates of the 
vertices A, B, C, and D then become 

x^=rsm6^, y^=-rcos6^, 

Xg=rcosd^, yg=rsin9^, 

X(^ =~rs'md^, Jc =rcos6^, 

Xu =-rcosd^, yD=-rsin6^ 



(3.46) 



a 



where r = —^ . From (3.46) 
V2 



x^+Xg+Xc+Xj,=0, 



(3.47) 



The complete set of coordinates of points A, B, C, and D are therefore 
A{x^ y^ h), B{x, y, h) , C{xc yc h) . D{x^ y^ h) (3.48) 

where h is the height of the moving square ABCD above the base square EFGH. 



-(v- -. ^ __ 



^§^ 



ex 



-t) 



1 m 


__^ ^"-^ 


u 


1^ III * ' 

\ 


h 




\ 


/' 






^T? 




.J! 


rl' 






/ /A^^ 


a: 


iTvJw 


\ 






^ \ 





Figure 3.8: Platform rotations about the z-axis 



39 



Then the coordinates for the corresponding lines $i, $2, . . ., $8 are given by 



5,= 



b b , bh bh b{x^-yj 

"2^2 22 2 



(3.49) 



S,= 



Xa — 



>'a+T' ^' -^T' - 



bh bh bix^ + y^) 



(3.50) 



S3 = 



b b bh bh b(Xg + yg) 



(3.51) 



5,= 



■^5 = 



^B 2' ^« 



-^c --' >'c 



6 


/i; 


bh 


fo/i 


2' 


2' 


2 


2' 


/i; 


2' 


2 



^Ub->'b) 



K^c-^c) 



(3.52) 



(3.53) 



^6 = 



b b , bh bh b(Xr + Vr ) 

^ 2 "^ 2 2 2 2 



(3.54) 



5,= 



V 4-^ V -.^ h- ^ ^ Hxp + yp) 

'2' ^^ 2' ' 2' 2' 2 



(3.55) 



S,= 



b b , bh bh b{Xry-yn) 

"2' ^"2' ' 2' 2' 2 



(3.56) 



It is apparent from Figure 3.8 that 1^=1^=1^=. l^ and 1^=1^=1^=1^. The 
corresponding Jacobian matrix can be determined by (3.11). Furthermore, calculating 
ydetJJ^ yields 



VdetJJ^ = 



4V2a'ft'/i^|cos/9, 






(3.57) 



where 



40 



/,=. 



l2 = 



42a sin 0+b 



V r 



42acose, -b^ 



+ h\ 



42asmd,-b^ (42acos0,-b ,, 

+ h 



With a-l,b = V2, and h = — , from (3.25) and (3.57), the quality index becomes 



X = 



cos 0, 



jy/detJJ^^ 

V'l^t J m J m (2 COS^ ^^ - 4 COS ^^ + 3)2 



(3.58) 



This is plotted in Figure 3.9 and it shows how the quality index varies as the platform is 
rotated about the vertical z-axis through its center. The eight legs are adjusted in length to 
keep the platform parallel to the base at a distance h. It is shown in the figure that the 
manipulator has the highest quality index A = 1 when ft = 0°, and A = (singularity) 
whenft=±90°. 



1.0- 

1 0.6- 

.^- 

3 0.4- 
O 

0.2 - 
0.0- 












/'^\ 






K 








y 




1 i i i ^ 



30 60 90 



Rotation Angle (degree) 



Figure 3.9: Quality index for platform rotation about the z-axis 



41 

It is interesting to note that the redundant 4-4 parallel manipulator shown in 
Figure 3.1 always becomes singular when its platform rotates ^^=±90° about the z-axis 

from its central symmetric position. This can be seen from (3.57), -^det JJ^ = when dz 

= ± 90°. Figure 3.10 illustrates the singularity position of the redundant 4-4 parallel 
manipulator when 6z = 90°. It is not immediately obvious from the figure why the eight 
connecting legs are in a singularity position. This kind of singularity has been discussed 
in detail by Hunt and McAree [14]. They explain that at such position, even when all 
eight leg actuators are locked, the connectivity between the base and moving platform is 
one. The moving platform can move instantaneously on a screw reciprocal^ to the eight 
leg forces on the z-axis with pitch h^, i.e., a screw with coordinates 

[O, 0, 1; 0, 0, h]. (3.59) 



H 


y{ 


I 


G 


V 




E 


^ 


SK \ 

--♦♦y^ \ \ 


X 

F 







Figure 3.10: Plan view of the singularity position of the redundant 4-4 in-parallel 

manipulator when 6^ = 90° 



When a wrench acts on a rigid body in such a way that it produces no work while the 
body is undergoing an infinitesimal twist, the two screws are said to be reciprocal. 



42 



Now when ^z=±90°, from (3.49) through (3.56), the component of moments 
about the z-axis for each of the eight legs are all the same -• 



A^.. = h and R, = ± 



■Jlab 



(/ = 1,2,...,8) 



(3.60) 



The coordinates for the eight legs become 



5.= 



L,, M,, h; P^, e,, ± 



-Jlab 



Hence, from (3.59) and (3.61), 



h,h± = or h,=+ 



(3.61) 



(3.62) 



4 ' Ah 

It follows that all eight legs lie on a linear complex, which is a three-parameter system of 
linearly dependent lines (Hunt [13]). 



CHAPTER 4 

THE KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 

4-4 IN-PARALLEL MANIPULATOR 

The kinematic analysis of in-parallel manipulators deals with the study of the 
platform motion determined by the leg displacements. Two problems can be 
distinguished for the kinematic aspects: inverse kinematics and forward kinematics. The 
inverse kinematics problem, i.e., finding the leg lengths for a given location (position and 
orientation) of the mobile platform (a difficult problem for serial manipulators), is 
straightforward for parallel manipulators. On the other hand, the forward kinematics 
problem, i.e., finding the platform location from a given set of leg lengths, is much more 
difficult. In general, this problem has more than one solufion for nonredundant cases. As 
an example, the forward analysis for the general 6-6 platform requires the solution of a 
40* degree polynomial (Raghavan [31]), the solution of which is clearly impractical for 
real-time implementation. A lot of methods have been presented to solve various types of 
nonredundant parallel manipulators as summarized by Dasgupta and Mruthyunjaya [7]. 

However, few works have been done on the kinematic analysis of redundant 
parallel manipulators, particularly the forward kinematics. A similar problem for 
determining a unique position and orientation of the platform of a general geometry 
parallel manipulator by using seven linear transducers has been solved by Innocenti [15]. 
He constructed a 146x147 constant matrix for solving the problem, which makes the 
computation time still larger than real time. Also, Innocenti's method produces only one 
solution for a general geometry parallel manipulator, but when the fixed base of a 



43 



44 

manipulator is in a plane, there should be two solutions instead of one since the 
manipulator can have two reflection configurations through the base according to the 
same set of readings from linear transducers no matter how many transducers are used. 
Therefore, his method may not be correct to solve the problem when the base of a parallel 
manipulator is in a plane. 

Zhang, Crane and Duffy [37] have performed the forward kinematic analysis on a 
special redundant parallel mechanism whose platform and base are connected by a ball- 
and-socket joint with four legs to determine the orientation of the platform. 

In this chapter the kinematic analysis of the redundant 4-4 in-parallel manipulator 
shown in Figure 3.1 is performed. From here on, the notations from Crane and Duffy [5] 
are used to describe the coordinates of point and the transformation matrix. For example, 
the notation Pa is used to indicate the coordinates of a point A as measured in a 
coordinate system I and ^R is defined as the orientation of the coordinate system 2 
relative to the coordinate system 1 . 

4. 1 Inverse Kinematic Analysis 

The objective of the inverse kinematic analysis of the redundant 4-4 parallel 
manipulator is to find the eight leg lengths for a given position and orientation of the 
square moving platform. 

In Figure 4.1, coordinate systems 1 and 2 have been attached to the fixed base and 
the moving platform respectively. The origins of the coordinate systems 1 and 2 are 
located at points £ and A receptively. The coordinates of points E, F, G, and H on the 
base are known in terms of the coordinate system 1 and 



45 










b 




b 







Pe = 





'p - 





'P - 


b 


'P = 

* H 


b 























(4.1) 



while the coordinates of points A, B, C, and D on the platform are known in terms of the 
coordinate system 2 and 



(4.2) 










a 




a 







Pa = 





^p - 





^P - 

* c 


a 


^P - 


a 
























Figure 4. 1 : Coordinate systems of a redundant 4-4 in-parallel manipulator 



46 



It is well known that the location of a rigid body in space can be described by the 
position and orientation of a coordinate system attached to the body with respect to a 
fixed reference frame. Thus, for the inverse analysis, the position and orientation of the 
coordinate system 2 is considered known, and can be given by the position vector ' P^ , 
which defines the position of the origin A of the system 2 relative to the origin E of the 
fixed frame 1, and a rotation matrix jR, which is a 3x3 matrix whose columns are the 
unit vectors along the coordinate axes of the system 2 as measured in the system 1 . Then 
the coordinates of points B, C, and D in terms of the coordinate system 1 become 



Pc - Pa + 2^ Pc 



'P, = 'P, + ]R^P,. 



Finally, the eight leg lengths can be calculated by 

=V('Pa-'Pe)-('Pa-'Pe). 
=V('Pa-'Pp)-('Pa-'Pf). 
= V('Pb-'Pf)-('Pb-'Pf). 
=V('Pb-'Pg)-('Pb-'Pg). 
=V('Pe-'Po)-('Pc-'PG). 
=>/('Pc-'Ph)-('Pc-'Ph). 
7=V('Pd-'Ph)-('Pd-'Ph). 
=V('Pd-'Pe)-('Pd-'Pe). 



(4.3) 



(4.4) 



where ■ represents the dot product of two vectors. 



47 

Hence, for a given location of the moving platform, there is only one possible 
solution for each leg length. 

4.2 Forward Kinematic Analysis 
The objective of the forward kinematic analysis of the redundant 4-4 parallel 
manipulator is to find the location of the moving platform given the actuator 
displacements of all the eight legs. Thus, the coordinates of points A, B, C, and D 
measured in the coordinate system 1 shown in Figure 4.1 need to be determined for a 
given platform side a, base side b, and eight leg lengths /, (/= 1, 2, ..., 8). 

4.2.1 Introduction 

The forward analysis is performed in detail in this section and thus provides a 
unique solution for the location of the moving platform above the base platform together 
with a reflected solution through the base for an arbitrarily specified set of eight leg 
lengths. However, extreme care must be taken in applying this analysis since what 
appears to be an arbitrary set of leg lengths may well be special and the solution will fail. 
For such cases, the constraint equations employed in the analysis presented here become 
linearly dependent in one way or another. 

A class of special cases has been reported by Selfridge [32] where he obtained a 
pair of assembly configurations (as opposed to a unique solution) above the base and a 
corresponding pair of reflected solutions through the base. It is interesting to note that 
one class of solutions reported by Selfridge [32] occurs when the platforms are parallel, 
the odd leg lengths are all equal and /, = /j = /, = /^ = / . Further, the even leg lengths are 
all equal and 1^=1^=1^=1^=1'. This class of solution embraces the workspace 
generated by a rotation of the top platform about the z-axis (Figure 3.8). While this does 



48 

not raise a problem with the quality index analysis it is important to recognize that these 
are in fact a pair of assembly configurations above the base platform. A numerical 
example is present in section 4.2.5. All this of course raises the issue of other classes of 
special cases that are worth further investigation. 

4.2.2 Coordinate Transformations 

First, the coordinates of any point in coordinate system 2 need to be transferred to 
coordinate system 1 . To do so, coordinate system 2 may be obtained by initially aligning 
it with coordinate system 1 and then introducing the following transformations: 

1 . Rotate the coordinate system 2 by an angle d\ about the ;c-axis until the >'-axis is in the 
plane defined by points A, E, and F, and the scalar product of the >'-axis with the 
vector Si is positive as shown in Figure 4.2. 

2. Translate the origin from point Eto F along the positive x-axis. 

3. Rotate by an angle ^ about its current z-axis, which causes the jc-axis to point along 
the vector S2. 

4. Translate the origin from F to A along the negative x-axis. 

5. Rotate the coordinate system about its current jc-axis by an angle 62 until the y-axis is 
in the plane defined by points A, F, and B, and the scalar product of the ^'-axis with 
the vector S3 is positive. 

6. Rotate by an angle ^ about its current z-axis, which causes the a:-axis to point along 
the vector S3. 

7. Rotate by an angle ^ about its current jr-axis until the y-axis points along the vector 
S4. 



49 




Figure 4.2: Coordinate transformations 



The coordinates of points A, B, C, and D may now be expressed in the coordinate 
system 1 as ?. ' 

'P3 = R,K+R3(t,+R,R,R/P3)], 

'Pc = r,[t,+R3(t,+r,r,r/pJ], 
'p, = r,[t,+R3(t,+R3R,r/pJ], 

where 



(4.5) 



50 





1 










b 






COS0, 


-sin0, 


0" 




'-h 


R,= 





cos^, 


-sin^, 


. T2 = 





» 


R3 = 


sin0, 


COS0, 





T = 










sin^, 


cos^, 

















1 









1 








- 


cos 02 


-sin 02 




'1 





R5 = 





cos ^2 


-sin ^2 


. R6 = 


sin 02 


cos 02 


. R7 = 





COS ^3 -sin ^3 







sin ^2 


cos ^2 












1 







sin ^3 COS ^3 



and the coordinates of points A, B, C, and D are known in terms of the coordinate system 
2 and are written as ^P^, ^Pg, ^P^, and ^P^. 

The angles 0i and 02 are shown in Figure 4.2 as the inner angles of the triangles 
AEF and AFB, respectively. Therefore, the angles 0i and 0^ are constrained to lie in the 
range of to K. The cosines of 0i and 02 may be determined from a planar cosine law as 



cos 01 = 



cos 02 



b'+ll-l^ 
2bU 



a^+ll-ll 



(4.6) 



laL 



and the values of 0i and 02 are determined as the inverse cosine value in the range of to 



n. 

The coordinates of points A, B, C, and D as measured in the coordinate system 1 
have been written as a function of the parameters di, 02, and ft. The objective now is to 
determine these parameters that will locate points A, B, C, and D such that they satisfy 
the distance constraints with points E, F, G, and H. 



51 

4.2.3 Constraint Equations 

Since the eight leg lengths, /, (/=1, 2, ..., 8), have been given for the forward 
analysis, the distance between points A, B, C, and D on the platform and points E, F, G, 
and H on the base must satisfy these leg lengths as shown in Figure 4.2. The coordinates 
of points A, B, C, D, E, F, G, and H all have been expressed in terms of the coordinate 
system 1 (see (4.5) and (4.1)), and the distance between these points may be expressed in 
the coordinate system 1 as 

('P3-'p^)■('P3-'p,)=/^ : (4.7) 

('P,-'P^)-('Pc-'Po) = /5\ (4.8) 

('Pc-'Ph)-('Pc-'Ph) = ^6. (4.9) 

(•P„-'Ph)-('P^-'Ph) = /^ (4.10) 

('P^-'P,)•('P„-•p,) = /^ (4.11) 

Note that three constraint equations are not written for the distance between points A and 
E, A and F, and B and F. The distance between these points will be equal to /i, I2, and I3, 
respectively. These three leg lengths have been used in the transformation of coordinate 
systems, which relates the coordinate systems 1 and 2 included rotation angles 0i and ^ 
and origin translation distance Ij. 

Equations (4.7) through (4.1 1) may be expanded and factored into the form 
labs^SiS^ -2bcj(as^c^c-^+s^(ac^ -l2))+a^ +b^ -^ac^l^ +ll - l] =0, (4.12) 

labs^ (s^ s^ + c^ s^Cj + c^s^ ) + Ibc, [ac^ s^s^ - ac^ s^c^ - a{c^ c^ s^ - s^ s^ )c^ - 



s^{ac^ -l2))+2a^+b^ -lac^l^ +ll -ij =0, 



(4.13) 



52 

s^ iac^ - ^2 ))+ 2aZ75^ (^2^3 -s^c^)- 2a(bs^ c^ c^ + (bc^ - 1^ )s^ )c^ + (4. 14) 

2a^ +2b'^ +2abc^c^ -2bc^l-^-2ac^l^+ll -ll =0, 

2abs^ (c^ s^c^ + c^s^ ) + 2bc^ [ac^ s^s^ - a(c^ c^^ c^ - s^ s^^ ^-^ + l^s^ )+ 
2abs^ s^s^ - 2a[bs^ c^ c^ + {bc^ - 1^ )s^ )c^+a +2b - 2bc^ l^+l^-lj =0, 

2abs^s^s^-2a{bs^c^c^ +bc^s^ -IjS^^Y^ +a^ +b^-2bc^l^ +ll -1^=0 (4.16) 

after recognizing that sf +cf =1 and s^ +c^ =1, and where Si, c„ (/= 1, 2, 3) represent 

the sine and cosine of di and s^ , c^ ,(j= 1, 2) represent the sine and cosine of ^. The 

objective now is to determine values for 61, &i, and ft which will simultaneously satisfy 
the five equations represented by (4. 1 2) through (4. 16). 

4.2.4 The Solution 

To solve equations (4.12) - (4.16), we consider s\, c\, S2, c^, 53, and C3 as 
independent variables. It gives us three more equations since 

sin ' d, + sin ' 6, = sf + cf = I, {i = 1, 2, 3). (4.17) 

Now equations (4.12) - (4.16) will be manipulated to eliminate ^1, c\, 53, and C3 first. The 

algebra to achieve this is what follows. 

Adding equations (4.12) and (4.15), and then subtracting (4.14) yields 

k^c^+k^C2+k^=0, (4.18) 

where ki, k2, and ^3 are known constants and are defined in Appendix A. Similarly, an 

equation that is linear in cj and C3 is generated by subtracting (4.14) from the sum of 

(4.13) and (4.16): 

k2C^+k^Cj+k^=0. (4.19) 



53 



Solving equations (4.18) and (4.19) for c\ and c^ respectively yields 



c. =-r-c2-r-. (4.20) 



^3 ~ ; ^2 ~ 



(4.21) 



k k 

Now substituting the expressions for c\ and C3 into equations (4.12) and (4.16) 
produces 

kf^s^S2 + k^c\ + /CgCj + /:, =0, (4.22) 

k^s^s-^ +k^Qsl+k^^c^^ +/:,2 =0. (4.23) 

Solving the above two equations for s\ and .^3 respectively yields 

^^^_ V2+V2+^9 ^ (4.24) 

"^6-^2 
„ _ ^10-^2 "^^11^2 "*""^12 /A^c\ 

Finally, substituting the expressions for ^i, 53, c\, and cj, into equations (4.15) and 
5,. + c,. =1 for i = 1 and 3 produces three equations in two variables, 52 and C2. Further, 

replacing ^2 with 1 - Cj , it is interesting to note that S2 cancels from these equations. This 
leaves following three equations in only one unknown, C2 

Eq, (C2 ) = X M,c^' = (/ = 1, 2, 3) (4.26) 

where the constants My are defined in Appendix A. The objective now is to determine 
value for C2 that simultaneously satisfy the three equations represented by (4.26). 



54 

Multiplying the three equations in (4.26) by C2, we obtain three additional 
equations. Thus, a total of six equations in the unknown C2 are obtained. These equations 
can be written in matrix form as 



My=0 



(4.27) 



where 



M= 



M,4 Af,3 M,2 M,, M,o 



^24 ^23 ^22 ^1\ ^2(i 



^34 -^33 ^32 ^31 ^30 ^ 



Af,4 M,3 M,2 M,, M,o 



M24 M23 M22 M21 Mjo 



M34 M33 M32 M3, M30J 



y = 



Here, we treat Cj , c^, Cj, Cj, Cj, and 1 as unknowns and thus equation (4.27) can be 
regarded as a homogeneous linear system in six unknowns. The trivial solution of y = 
is not feasible, since the last element of y must equal 1 . Solutions other than the trivial 
solution exist only if the homogeneous equations are linearly dependent, and as such the 
determinant of the matrix M must equal zero. Evaluating this determinant and seeing 
how close it is to zero will provide an indication of the quality of the measured data (i.e., 
the platform side a, base side b, and the joint positions) and the sensed data (i.e., the eight 
measured displacements /], /i, ..., k). The issue of how close to zero is satisfactory is not 
addressed in this dissertation. 

The six equations represented by (4.27) may now be rearranged into the form 

Ux = V (4.28) 



where 



55 



U = 



M,4 M,3 M,2 M,, M,o 



^24 ^23 ^^22 ^2\ ^20 



M34 M33 M32 M3, M30 



M„ M,3 M„ M„ 



Af24 M23 M22 M21 



M34 M33 Af32 M3, J 





■-2^" 










-2^ 







x = 


c,^ 


v = 




> 


C2^ 




-A^.o 
-M20 




^2] 




_-M3o_ 



Equation (4.28) represents six linear equations in five unknowns. The vector x may be 
solved for by selecting any five of those equations. The term cj is the fifth component of 
the vector x and unique value for this term is thereby determined. However, it should be 
noted that since &i is in the range of to In, there are actually two solutions of Si for a 
value of ci. Thus, the manipulator has two configurations for a given set of leg lengths. 
These two configurations are due to a reflection through the base plane. 

For each value of &i, corresponding values for c\, cs, s\, and 53 can be calculated 
from (4.20), (4.21), (4.24), and (4.25) respecfively. Then, values for 0i and <% can be 
determined. Finally, the coordinates of points A, B, C, and D in terms of the coordinate 
system 1 can be obtained by substituting 6[, 62 and 03 into (4.5). 

4.2.5 Numerical Verification 

In this section, a numerical example is presented for a redundant 4-4 parallel 
manipulator to verify the analysis. The dimensions of the manipulator are measured in an 
arbitrary length unit and given as follows: 

platform side a = 10, base side b=l5. 
A set of leg lengths are given as 



56 

/, =13.62421, /^ =10.40411, /j =14.47201, 7^=1 1.16409, 
/, =16.34095, /^ =17.59696, /^ =16.22984, /g =15.92500. 

The numerical results are presented in Table 4. 1 and two configurations are shown to be 
reflecting through the base plane. Thus, a unique configuration may be easily determined 
by checking the sign of z coordinate of one of the platform joints. In order to verify these 
results, an inverse kinematic analysis was performed. All solutions reproduced the correct 
leg lengths. 



Table 4. 1 : Numerical results of the redundant 4-4 in-parallel manipulator 



No. 


^1 (deg.) 


Oi (deg.) 


Si (deg.) 


Pa 


'P 


'Pc 


■p„ 


1 


-105.534 


133.523 


-27.872 




"10.079" 

2.455 
8.832 






"16.119" 
10.327 
10.077 






" 8.921" 
15.045 
15.168 






" 2.881] 

7.173 
_13.923j 


2 


105.534 


-133.523 


27.872 


BSS 


'10.079" 

2.455 

-8.832 






" 16.119" 

10.327 

-10.077 






8.921' 
15.045 
-15.168 






2.881] 
7.173 
-13.923] 



In the following example the above solution failed because equation (4.7) through 
(4.11) become linearly dependent while equation (4.9) through (4.11) are redundant for 
the system. Using the same dimensions as the above example, the leg lengths now 
become: 



^1=^3=^5=^7=^ = 18 and l^=l^=l^=l^=r=\e. 



57 

The numerical results are presented in Table 4.2. It is apparent that there are now 
two configurations above the base plane and further two solutions reflected through the 
base. 



Table 4.2: A numerical example for the special case of 
the redundant 4-4 in-parallel manipulator 



No. 


h 


K 


P 


'Pc 


P 








'2.267" 






'-6.698' 






"-2.267" 






" 6.698" 




1 


5.199 




6.698 
5.199 






2.267 
5.199_ 






-6.698 
. 5199 






-2.267 
5.199 










■ 2.267' 






' 6.698' 






"-2.267" 






"-6.698" 




2 


15.099 




-6.698 
15.099 






2.267 
15.099 






6.698 
15.099 






- 2.267 
15.099 










" 2.267' 






'-6.698' 






"-2.267" 






" 6.698" 




3 


-5.199 




6.698 
-5.199 






2.267 
-5.199 






-6.698 
-5.199 






-2.267 
-5.199 








r 


2.267 ■ 






6.698" 






"-2.267" 






-6.698" 




4 


-15.099 




-6.698 






2.267 






6.698 






-2.267 








■ 


-15.099 






-15.099 






-15.099 






-15.099 





CHAPTER 5 

THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

4-8 IN-PARALLEL MANIPULATOR 

A redundant 4-8 parallel manipulator is shown in Figure 5.1, which is derived 
simply by separating the double ball-and-socket joints in the base of the redundant 4-4 
manipulator shown in Figure 3.1. A plan view of the redundant 4-8 parallel manipulator 
is shown in Figure 5.2. The device has a square platform of side a and an octagonal base 
formed by 4 pairs of joints Ed and Ea, Fa and Fb, Gb and Gc, and He and Hq. Each pair 
of joints is separated from a vertex of a square of side t by a distance fib for which 

0</3<— . Clearly the platform is degenerate when fi= —. 




Figure 5. 1 : A redundant 4-8 in-parallel Figure 5.2: Plan view of the redundant 4-8 
manipulator in-parallel manipulator 



58 



59 



5.1 Determination of ^det J^ J 



The moving platform of the redundant 4-8 parallel manipulator shown in Figure 
5.2 is located at its central symmetrical configuration and is parallel to the base with a 
distance h. By analogy with the redundant 4-4 parallel manipulator, a maximum value of 
the square root of the determinant of the product of the manipulator Jacobian by its 

transpose, i.e., ■^'^et J^jj^ , may be obtained for this configuration. 

A fixed coordinate system is placed at the center of the octagonal base as shown 
in Figure 5.2. Then, the coordinates of the points A, B, C, and D on the platform are 



-^ h 



, B 



} 



4ia 



h 



\ ( 
, C 



^ h 



, D 



42a 



h 



(5.1) 



The coordinates of the points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are 



r 



Gc 



( 



-^4 -- 

^ 2 



. F, 






d, -- 



■d, 



V 



. G, 



\ ^ J 



\ ( 

, Hf, 






, Hu 



d, 






(h 


^ 






d, 




\2 


; 


h 


^ 


~2 


-^4 C 


) 

J 



(5.2) 



where 



^4 = 



{l-2j3)b 



Counting the 2x2 determinants of the various arrays of the joins of the pairs of points 
EaA, FaA, .... EdD yields the Pliicker line coordinates of the eight leg lines. That is, from 
points Ea and A, 



5,= 



, V2a-ft , bh ^ , Sad. 
d„ — , h; - — , d,h, —^ 



(5.3) 



From points Fa and A, 



60 



S,= 



, -Jla-b , bh , , -Jlad^ 
-d„ — , h; -—, -d^h, ^ 



(5.4) 



From points Fb and B, 

Sa-b 



S,= 



^ u J u ^^ 42ad^ 
, d^, h; -d^h, — — , — -— 

2 2 2 



(5.5) 



From points Gb and B, 

\[ia-b 



S,= 



, , , , bh -Jlad, 

, -J4, h; d^h, — — , -— 

2 2 2 



(5.6) 



From points Gc and C, 



^5 = 



-d,. 



4ia-b 



, bh 



-d^h. 



42ad^ 



(5.7) 



From points He and C, 



k = 


, yfla-b , bh , . ^l2ad^ 
d„ ^ ' ^' 2 ' ' ^ 


■ 


From points Ho and D, 


S,= 


42a-b ^ , ^ , bh 42ad 
2 ' ^4. h; d,h, ^ , ^ 


i_ 



(5.8) 



(5.9) 



From points Ep and D, 



S,= 



42a-b 



, J4, h; -d^h. 



bh slad^ 



(5.10) 



The normalized Jacobian matrix of the eight leg lines can be expressed in the 



form 



J = 



\sj 


si 


si 


si 


si 


•^6 


si 




['■ 


h 


h 


U 


h 


h 


l. 


k 



(5.11) 



61 



where h, k, .■.,k are the leg lengths and since the device is in a symmetrical position, I] 
= l2 = ••■ = /g = /, and 



l = ^|L^+M^+N^ =J-[a^ -42ab + {2j3^ -2j3 + l)b^ +2h^]. 



(5.12) 



From (5.3) to (5.10), (5.11) becomes 



d^ -d^ d^ 



di -d^ d^ 



-dr 



-d, 





-d, 


-d, 


d. 


-d. 


d, 


d, 


-d. 




h 


h 


h 


h 


h 


h 


h 


-7 


bh 

2 


bh 
2 


-d^h 


d^h 


bh 
2 


bh 
2 


d^h 




d,h 


-d,h 


bh 

2 


bh 

2 


-d,h 


d,h 


bh 

2 




42ad, 


■sliad. 


42ad, 


42ad, 


Sad, 


Sad, 


Sad, 



-d,h 

bh 

2 
Sad, 



(5.13) 



where same as (3.13), d^ = 



Sa-b 



A{dl+dl) 



detJJ' = 



From equation (5.13), the determinant of the product J J becomes 

2{2dl-bd,)h 

4(rf'+^4) 2{bd,-2dl)h 

8/i' 

2{bd,-2dl)h (b^+4dl)h^ 

2i2dl-bd,)h (b^+4d^)h^ 





T 1 



/'^ 







4a' d 



2 J2 



(5.14) 



Expanding (5.14) and using (5.12), then extracting the square root, we obtain 



62 

VdetJJ =-r — ^ yr. (5 15) 

y -ynab + {2fi^-2p + l)b'+2h'\ 

Assuming the top platform size a is given and taking the partial derivative of (5.15) with 
respect to h and b respectively and equating to zero yield 

9642{l-2/3ya^b^h^[a^-yf2ab + (2fi^-2fi + \)b^-2hA ^* 

r 7^ T4 ~^ (5.16) 

[a^-yl2ab + {2fi^-2fi + l)b^+2h^\ 

and ' ^ * ' I ^ 

96yf2{-\ + 2fi)'a'b'h'[-a' +(2fi' -2/3 + \)b' -2h'] _^ 

\ t— i4 ~ (5-17) 

y-42ab + {2P^-2p + \)b^+2h^ 

Note that we already assumed fi^ — , then when a, b, and h are not equal to zero, from 

equations (5.16) and (5.17), we get 

a'-42ab + {2p^-2p + \)b^-2h^ =0, (5.18) 

-a^+{2p^-2p-\-\)b^-2h' =Q. (5.19) 

Subtracting (5.19) from (5.18) yields 

2a^-y[2ab = 0, (5.20) 

and thus 

b = ->l2a. (5.21) 

Substituting (5.21) into (5.19) gives 

(4y9'-4yff + l)fl'-2/i' =0. (5.22) 

The above equation yields two solutions for h, here we only take the positive solution 

h = -={\-2P)a. • (5.23) 



63 

Therefore, when b = -Jla and h = -7=(1 - 2fi)a , the redundant 4-8 parallel manipulator 

is at the maximum quality index as shown in Figure 5.3, and from (5.15) 

^JMJl = (VdetJJ^)_ = 4yf2a' (5.24) 

where Jm denotes the Jacobian matrix of this configuration. It is interesting to note that 
this maximum value of Vdet JJ ^ is independent to the value of fi. 




Figure 5.3: Plan view of the optimal configuration of the redundant 4-8 in-parallel 
manipulator with the maximum quality index 



From (3.24) and (5.24), it shows that both the redundant 4-4 and the 4-8 parallel 



manipulators have the same maximum value of VdetJJ^ . Figure 5.4 illustrates the 
compatibility of these two results. It can be observed that as the distance between the 
pairs of separation points of the double ball-and-socket joints E, F, G and H of the 
original 4-4 manipulator increases, the height h at which the manipulator has the 



64 



maximum quality index decreases (see (5.23)) from h = —i= {P= 0, concentric ball-and- 

V2 



socket joints) toh = 0(/3= —, platform is degenerate). 




Figure 5.4: Compatibility between the redundant 4-4 and the 4-8 parallel manipulators 

^ - V2 ^ 



5.2 Implementation 
From (1.2) and (5.24), the quality index for the redundant 4-8 parallel manipulator 
shown in Figure 5.1 can be expressed as 

VdetJJ^ ■• ■ 'I 



/l=- 



4V2fl- 



(5.25) 



In this section, a redundant 4-8 parallel manipulator with a = 1 and b = 42 \s, taken as an 
example for the investigation of the variation of the quality index X during a number of 
platform movements. 



65 

The first platform movement we studied is a pure vertical translation of the 
platform along the z-axis that passes through the center of the platform. From (5.15) and 
(5.25), the quality index for this movement becomes 



X = - 



[a^ -yl2ab + {2/3^ -2j3 + l)b^ +2h^\ 
With a = 1 and b = V2 , this reduces to 

16V2(l-2y9)'/i^ 



(5.26) 



X = 



[i\-2J3f +2h^Y 



(5.27) 



and is plotted in Figure 5.5(a) as a function of h and fi. Figure 5.5(b) plots the variation 
curves of the quality index for several different values of J3. From these figures, we can 
see the height (hm) at which the manipulator has the maximum quality index is reduced as 
fi increases. Each value of fl designates the distance between the separation points in the 
base and is a first design parameter. Clearly, fi=0 is the best overall design. 




(a) (b) 

Figure 5.5: Quality index for platform vertical movement 






66 



The second platform movement is a pure horizontal translation of the platform 
away from its initial location at height h. To derive an expression for -^detJJ^ , we 

assume the center of the platform move to point (x, y, h), then the coordinates of the 
points A, B, C, and D on the platform become 



X y- 



Sa 



h 



( 



4.^« U 

X y^ h 



B 



D 



x + 



42a 



2 
r V2a 



X-- 



y h 



y h 



(5.28) 



The coordinates of points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the octagonal base can 
be found from (5.2). Calculating the Plucker line coordinates for each of the eight leg 
lines yields 



5, = 



, , Sa-b , bh ^ , bx + djSa-ly) 



(5.29) 



S,= 



. Sa-b , bh ^ , bx-dJ-Jla-ly) 
x-d^, y , h; - — , -d^h, *- ^ 



(5.30) 



S3 = 



„, -Ha-b ^ , . . . bh d,(42a + 2x) + by 

x + — - — , y + d^, h; -d,h, - — , -^ — - — ^ 

2 2 2 



(5.31) 



"^4 = 



^Sa-b , , , , bh d.(42a + 2x)-by 

x + — - — , y-d^, h; d^h, - — , — *- — '- — ^ 

2 2 2 



(5.32) 



Ss = 



, ^y[2a-b J bh ^, bx-dA42a + 2y) 
x-d^, y + — , h; — , -d^h, i^^ ^ 



(5.33) 



^6 = 



^ , ^42a-b , bh ^ , bx + dAyl2a + 2y) 
x + d^, y + , h; —, d^h, ^ ^ 



(5.34) 



67 



S,= 



Sa-b , , ^ , bh d,{42a-2x)-by 

X , y-d., h; d.h, — , 

2 2 2 



(5.35) 



■^8 = 



42a-b ^ u ^^ 

X — , y + d^, h; -d^h, —, 



d^ (V2a - 2jc) + by 



(5.36) 



where same as (5.2), d^ 



i\-2j3)b 



The Jacobian matrix can then be constructed by using (5.11). Further, ^det JJ^ 



becomes 



j^-—^ a'b'h'jX-ipf^ljl^ +l^+l^+l^)(l', +ll +ll +ll) 



(5.37) 



where the leg lengths are 



l,=J{x + dJ + 




+ h\ l,=J{x-dJ + 



h=- 



x + - 



42a-b 




+ h\ 



+ {y + dj+h\ l,=. 



x + - 



Sa-b 



\2 



+ {y-dj+h\ 



ls=J{x-dJ + 




ln=- 



X — - 



^a-b 



l,=Mx + dJ + 




+ {y-dj+h\ /,=. 



X-- 



yfla-b 



2 , 1.2 



+ {y + dj+h 



With a = 1, Z7 = V2, and /i = /z„ = ^^ p^^ = ^-^ , from (5.25) and (5.37), the quality 



index becomes 



68 



, JdetJJ^ (1 - 2B)\l^ + ll + 1} + ll ) 

^= I, , ,, = ,,2,2,2,2 (5.38) 



where 



/, =le =4x^ +42{\-2l3)x+y^ +{\-2p)\ 

12=15= V^' -S(l-2j3)x+ y^+(l-2fi)\ 

h=k= V^' + V2(l -2l3)y + y^+{\-2p)\ 

1,=It =^x' -yf2{l-2j3)y + y' +a-2J3)\ 

In Figure 5.6, the quaUty index and its contours as the platform is translated away 

from the central location while remaining parallel to the base at h^, are drawn for various 

values of y9. It should be noted that when fi=0, the 4-8 manipulator becomes the 4-4 

manipulator and its corresponding quality index is drawn in Figure 3.5. Comparing 

Figure 5.6(a)-(d) and Figure 3.5, it is clear that the smaller fi, the larger workspace area of 

the platform is with high quality index. 

Now we attach a new coordinate system x'y'z' to the square platform. This 
coordinate system may be obtained from the platform configuration shown in Figure 5.2 
by initially aligning it with the xyz coordinate system on the base and then raising it by a 
distance h along the z-axis to the top platform. Thus, the x'- and >''-axes are parallel to the 
X- and y-axes respectively when the platform locates at its initial central position shown 
in Figure 5.2. 

We are interested in deriving -,/detJJ^ when the platform rotates about the x'- 
and y'-axes from its central position. Here, we only derive the platform rotation about the 



69 

y'-axis. But the result to be derived is the same for the platform rotation about the x'-axis 
since the redundant 4-8 parallel manipulator is fully symmetric. 




1.0- 



0.5 



>.0.0 



-0.5 - 



-1.0 



1 -1 



(a) ;& = -at ;i„= — -0.14 
'^ 5 ""10 




-1.0 -05 0.0 0.5 1.0 

X 




1.0 - 



0.5 - 



?^0.0 - 



-0.5 



-1.0 - 



1 -1 




1 J? 

(b) I5 = - at /i„ = ^^ = 0.24 

3 " 6 



70 




1.0 - 



0.5 



P^O.O 



-0.5 



-1.0 



1 -1 




4 4 




1.0 - 



0.5 



PnO.O 



-0.5 - 



■1.0 




T r 



1 -1 



■1.0 -0.5 0.0 0.5 1.0 



(d) fi = - aih= — ^0.53 
^8 "■ 8 



Figure 5.6: Quality index for platform horizontal translation with different values of fi 



i^ ■ " "'? T 5 



f: *■• 






71 



Figure 5.7 illustrates a side view of the moving platform ABCD rotated dy about 
the _y'-axis. The coordinates of the vertices A, B, C, and D become 







-Jla 



^ H 



\ 


/ 


h 


B 


J 


V 


\ 


/ 


t 


D 


> 


V 



■v2(3 -v2a 
cos^„ h-\ sin^„ 

2 • 2 ' 



\ 



cos^^ h sm^, 

2 ' 2 ' 



(5.39) 



and the coordinates of vertices Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base can be 
found from (5.2). 




Fb(Gb) 



Figure 5.7: Platform rotation about the ^^ '-axis 



Note that the positions of Une $i, $2, $5, and $6 do not change during this platform 
rotation and their corresponding Plucker line coordinates can be obtained from (5.3), 
(5.4), (5.7), and (5.8) respectively. The Plucker coordinates for the line $3, $4, $7, and $8 
are now given by 



72 



53 = 



^!la COS e^ - b 42a sin 9^ + 2h 

2 2 

d^ {42a sin 6^ + 2/i) b{42a sin ^, + 2/z) 42ad^ cos ^, 



(5.40) 



^4 = 



42a cos ^^, - ft 42a sin 6'^ + 2ft 



d^ {42a sin 6'^. + 2h) b{42a sin 6*^ + 2h) 42ad^ cos ^^ 



(5.41) 



5,= 



V2fl cos ^^. - b 



, -d^, -■ 



42a sin 6^ - 2h 



2 ^ 2 

d^ {42a sin 6^. - 2h) b{42a sin 0^ - 2h) 42ad^ cos d^. 



(5.42) 



5« = 



v2a cos ^y - £> 42a sin ^^ - 2/i 

~ > ^4> T > 



d^ {42a sin ^^, - 2h) b{42a sin <9^ - 2h) 42ad^ cos ^ 



(5.43) 



with d^ = 



(l-2y^)fe 



From Figure 5.7, we have li =1^ =1^ =1^=1 , 1^=1^, and l-j=l^. The 
corresponding Jacobian matrix can now be obtained by (5.1 1), and further -^det JJ^ can 

be determined. With a = \,b = V2, and h = h^= — iJ- , from (5.25), the quality index 

42 



then becomes 



73 



cos 30^ ) + (16y3' - 32/3' + lAp^ -Sfi- 2)icos20, + 1)) + (Z' + /' )(14y9^ 
6/? + l + 8y5'(y5-2)(cos2^y +l) + 2y3(5y9-l)cos2^y) + (/3' -l,)(S/3' - 

I2j3^+6J3-I)sin2ey2 



(5.44) 



where 



l = l-2J3, 



Zj = ^4y9^ - 4;5 + 2 - cos ^,, - {2/5 - 1) sin 6^ , 
/7=^4/?'-4y5 + 2-cos^^+(2y0-l)sin^^. 

Figure 5.8(a) plots the quality index as a function of ^ and /3. Figure 5.10(b) 
presents the change of the quality index for several different values of fi. 




(a) 



90 -60 -30 30 60 
Rotation Angle 6 (degree) 

(b) 



Figure 5.8: Quality index for platform rotations about the x'- and >''-axes 



74 

A plan view of the redundant 4-8 parallel manipulator with the moving platform 
ABCD rotated 6z about the z-axis is shown in Figure 5.9. The x and y coordinates of the 



vertices A,B,C, and D become 

x^=r sin 6^ , 
Xg = r cos 6^ , 
X(; =-rsin0^, 
x^ =-rcos0^. 



y^=-rcose^, 
yg =rsind^, 

y^=-rsme^ 



(5.45) 



where r = 



V^ 



The complete set of coordinates of points A,B,C, and D are therefore 
M^A Va h), B{xg yg h), C{xc yc h) , D{xj, y,, h) 
where h is the height of the moving square platform above the octagonal base. 



(5.46) 



H 

Hd 


ifc 


y 




=^ 


Gb 


/3 

/ ^ 


- - .^ _^ 


r 








1 ^vT 


^ 




i 




T,.^4 


l/\ 


X 


-t 


1 yiJ^^ 




rt\W 


\ 






■n 


1/ 


\ 


^ \ f 

A"" - - -. ^ 


^/ 


Fb 
F 


E 


Ea 


1? 


Fa 



Figure 5.9: Platform rotation about the z-axis 



75 



The coordinates for the corresponding lines $i, $2, ..., $8 are then given by 



S^ = 



x,+d^, yA+—, h; — — , d^h. 



bix,-y,+2fiy,) 



(5.47) 



5,= 



b 



bh 



^A-d^, yA+TT' h'^ — —^ -d^h. 



bix, + y,-2j3y,) 



(5.48) 



S,= 



b _, , . , . bh b{xg-2fixg+yg) 
^B--' ya+d^, h; -d^h, - — , ^^ 



(5.49) 



^4 = 



b A u A u bh bjxg-ipxg-yg) 
^B--' yB-d^, h; d,h, -—, 



(5.50) 



^5 = 



b , bh b{Xc-yc+2J3yc) 

^c-^4. yc~2' ' Y' "^ ' 2 



(5.51) 



Se = 



b . bh . . b{Xc + yc~ '^fiyc ) 



Xc+d^, yc-T' ^; Y' d^h, - 



(5.52) 



5, 



b , , . , bh 

^D + 2' ^o~^4' h- d,h, —., 



biXa-iPx^ + y^,) 



(5.53) 



S,= 



V ^^ „ -i-w A- ^h ^^ b{x^-ipx^-yi,) 
^D + 2' ^o+«4. h\ -d^h, — , 



(5.54) 



From Figure 5.9, we have l^ =1^ =1^ = /, and l2=h -h-h- ^^e corresponding 



Jacobian matrix can now be determined by (5.1 1), and further ^det JJ^ becomes 

I 4V2aV/i'(l-2/?)^|cos6> I 

VdetJJ^= ) ^' ' ^ (5.55) 



where 



/, =-^l(^[2asme^ +b-2fibf +^2000^8^ -bf +4h^ , 



k =-i(42asme^ -b + 2fibf +y2acose^ -bf +4h' 



76 



1 — 2/? 
With a = \, b = yfl, and h = h„= — ^ , from (5.25) and (5.55), the quality index 

V2 



becomes 



A = - 



(l-2/?)'|cos^^ 



3 • 

(2(2y5' -2yff + l)(cos^^ -2)cos^^ +16/?' -32y9' +28y3' -12y9 + 3> 



(5.56) 



This is plotted in Figure 5.10(a) as a function of yff and 6^. It shows how the quality index 
varies as the platform is rotated about the vertical z-axis through its center. The eight legs 
are adjusted in length to keep the platform parallel to the base at a distance /im- Figure 
5.10(b) illustrates the variation of the quality index for several different values of fi. It is 
shown in these figures that the manipulator has the highest quality index A, = I when 
6z=0°, and X = (singularity) when 0z=± 90°. As can be seen in Figure 5.10(b), a slight 
change of 0z under a large fi has a much greater impact on the quality index than that of 
the same change under small fi. 




1.0 



0.8 



X 



■D 0.6 

c 



§ 0.4 
O 



0.2 







A 





0.0 

-90 -60 -30 

Rotation Angle 6 (degree) 
(a) (b) 

Figure 5.10: Quality index for platform rotation about the z-axis 



90 



77 

Again, from Figures 5.8 and 5.10 we can see clearly that better designs are 
obtained as fi reduces to zero. Hence the best 4-8 parallel manipulator design is obtained 
when the pair of base joints are as close as possible. 



Since from (5.55), ^/detjj^=0 when ^^=±90°, a redundant 4-8 parallel 

manipulator always becomes singular when its platform rotates ^^=±90° about z-axis 
from its central symmetric position. Figure 5.11 illustrates the singularity position when 
a=90°. 




Figure 5.11: Plan view of the singularity position of the redundant 4-8 in-parallel 

manipulator when dz = 90° 



In complete analogy with the redundant 4-4 parallel manipulator presented in 
Chapter 3, when ^^=±90°, the moving platform of the redundant 4-8 parallel manipulator 
can move instantaneously on a screw reciprocal to the eight leg forces on the z-axis with 



pitch h = + 



-Jlab 
Ah 



This is because for ^z=±90°, from (5.47) through (5.54), the 



78 
component of moments about the z-axis for each of the eight legs all are equal to 



CHAPTER 6 

THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 

4-8 IN-PARALLEL MANIPULATOR 

The kinematic analysis of the redundant 4-4 parallel manipulator has been 

performed in Chapter 4. It is shown that the inverse kinematics is straightforward for 

parallel manipulators while the forward kinematics is difficult. In this chapter the forward 

kinematics of the redundant 4-8 parallel manipulator shown in Figure 5.1 is studied. It 

will be shown how this problem can be easily solved by transferring it to the 

corresponding redundant 4-4 case which then can be solved by using the method 

presented in Chapter 4. 

6. 1 Forward Kinematic Analysis 

In Figure 6. 1 , coordinate systems 1 and 2 have been attached to the fixed base and 
the moving platform of a redundant 4-8 parallel manipulator, respectively. The origins of 
the coordinate systems 1 and 2 are located at points E and A receptively. The objective of 
the forward kinematic analysis of the redundant 4-8 parallel manipulator is to find the 
position and orientation of the moving platform given all the eight leg lengths. 

Eight dash lines connecting the moving platform and the base are drawn in Figure 
6. 1. K we connect the platform with the base by legs along these dash lines to replace the 
original legs, we obtain a redundant 4-4 parallel manipulator with the platform location 
same as the original 4-8 manipulator. Thus, if we can determine the distances, /o, (/= 1, 2, 
..., 8), between the platform and the base along the dash lines, the forward kinematic 
problem will have been solved by using the method presented in Chapter 4 for the 



79 



80 

redundant 4-4 parallel manipulator. The objective now is to determine values for Iq\, Iq2, 
. . . , /o8 from the original leg lengths li,l2,...,k. 




Figure 6. 1 : Coordinate systems of a redundant 4-8 in-parallel manipulator 



Observing Figure 6.1, we find l\, h, hi, and /02 are in the same plane defined by 
points A, Ea, and Fa while Ij, U, /03, and /04 are in the plane defined by points B, Fb, and 
Gb, I5, k, hs, and /06 are in the plane defined by points C, Gc, and Hq, and /?, k, /07, and /qs 
are in the plane defined by points D, Ho, and Fd. Thus, /q/ can be determined from /, in 
the same plane. For example, in the plane defined by points A, Fa, and Fa as shown in 
Figure 6.2, we have 



W^' 



:'-lf\.: ■' . : ;■ ^' 



81 



/o, = 4P^b^ + 1^ - '^fibl^ cos(;r -?),), 

/o2 =^(l-fi)'b'+lf-2(\-/3)bl,cos(p,. 



(6.1) 
(6.2) 



where 



cos 9), = 



2^(1 -2y9)/, 




Figure 6.2: Leg relations 



Similarly, the other leg lengths can be obtained 

/o3 = Vi^VT/^^^2^W^"cos(^^^^, 

/o5 = -^P^b"- + /j' - Ipbl, cos(;r - 9^3 ) , 



(6.3) 
(6.4) 
(6.5) 



82 



loe = yl (^ - fi)' b'+l's- 2(1 -/3)bl, cos (p,, (6.6) 



/o7 =V^V+/f^^2y9w7cos(^^^^, (6.7) 

l^= 4{^- Pfb^ +It -2{\- p)bl, coscp, , (6.8) 



where 



(1- 


-iPfb'+ll-ll 




2b{\-2fi)l, 


_o- 


-2Pfb'+ll-ll 




2b {\- 2 15)1, 


(1- 

nr\e m — 


-2pfb'^l^-ll 



2b {\- 2 15)1, 

Using the values of ki, I02, ..., /os as input leg lengths to the forward kinematic 
analysis presented in Chapter 4, the position and orientation of the moving platform will 
be determined. 

6.2 Numerical Verification 
A numerical example is presented for a redundant 4-8 parallel manipulator to 
verify the analysis. The dimensions of the manipulator are measured in an arbitrary 
length unit and given as follows: 

platform side a = 10, base side b= 15, B =—. 

8 

A set of leg lengths are given as 

/, =12.21787, /^ =9.15596, /, =12.83105, /^ =7.52035, 

/, =13.47917, /6= 13.13367, /,= 13.88865, /« =14.04687. 



83 

Thus, the input leg lengths for the forward analysis of the corresponding redundant 4-4 

parallel manipulator are obtained from (6. 1) through (6.8) and 

/o, = 13.59387, /o2 = 9.87590, l^, = 9.87590, l^^ = 7.94680, 
/o5 = 14.41631, /o6 = 13.98464, l^^ = 14.72302, l^s = 14.92181. 

The numerical results are presented in Table 6.1. It has been verified by an inverse 
kinematic analysis that all solutions reproduced the correct leg lengths. 



Table 6.1: Numerical results of the redundant 4-8 in-parallel manipulator 



No. 


0\ (deg.) 


O2 (deg.) 


0i (deg.) 


*Pa 


'P 


'P 


'Po 


1 


-112.939 


122.570 


-28.524 




"10.409" 
3.408 
_ 8.052 






"14.940" 

12.304 

. "7-475 _ 






' 7.091" 
16.592 
11.948 






" 2.560] 

7.696 

12.525] 


2 


112.939 


-122.570 


28.524 




'10.409" 

3.408 

-8.052 






"14.940' 

12.304 

_-7.475_ 






7.091' 
16.592 
-11.948 






2.560] 
7.696 



Similar to the redundant 4-4 parallel manipulator, there is a special solution when 
the platform rotates about the z-axis (Figure 5.9). For example, when the leg lengths now 
become 

/, =/3 =/5 =/^ =/ = 18 and 1^=1^ =1^=1^ =l'= 16 
for the redundant 4-8 parallel manipulator in the first example, two configurations above 
the base plane with another two reflected through the base are obtained as shown in Table 
6.2. 



84 



Table 6.2: A numerical example for the special case of 
the redundant 4-8 in-parallel manipulator 



No. 


h 


Pa 


'P 


P 




P 








■3.022" 






■-6.393' 






"-3.022" 






" 6.393' 




1 


7.498 




6.393 
7.498 






3.022 






-6.393 
7.498 






-3.022 
_ 7.498 










" 3.022" 






' 6.393' 






"- 3.022" 






"-6.393" 




2 


15.748 




-6.393 

_15.748_ 






3.022 
_15.748_ 






6.393 

15.748 






-3.022 
_15.748_ 










" 3.022' 






"-6.393" 






"- 3.022" 






" 6.393' 




3 


-7.498 




6.393 
-7.498_ 






3.022 
-7.498 






-6.393 

-7.498 






-3.022 
-7.498 










3.022' 


1 




6.393" 


1 




■ -3.022" 






" -6.393" 




4 


-15.748 




-6.393 






3.022 






6.393 






-3.022 










-15.748 


■ 




-15.748 






-15.748 






-15.748 








CHAPTER 7 

THE OPTIMUM QUALITY INDEX FOR A SPATIAL REDUNDANT 

8-8 IN-PARALLEL MANIPULATOR 

A redundant 8-8 in-parallel manipulator is shown in Figure 7.1, which is derived 
by separating the double ball-and-socket joints in the base and the top platform of a 4-4 
manipulator shown in Figure 3.1. A plan view of the redundant 8-8 in-parallel 
manipulator is shown in Figure 7.2. The device has eight legs connecting an octagonal 
platform and a similar octagonal base. The octagonal top platform is formed by 4 pairs of 
joints Ai and A2, fii and B2, C\ and C2, and D\ and D2. Each pair of joints is separated 

from a vertex of a square of side a by a distance cea for which 0<«r<— . Similarly, the 

octagonal base is formed by 4 pairs of joints En and £a, F/^ and Fb, Gb and Gq, Hq and 
Hd, and each of them is separated from a vertex of a square of side fc by a distance fib for 

which 0<P<—. This design has the distinct advantage that it completely avoids the 

mechanical interference problem associated with the design of double spherical joints. 

7.1 Determination of ydet.T^.T^ 



' m 



The moving platform of the redundant 8-8 parallel manipulator shown in Figure 
7.2 is located at its central symmetrical configuration and is parallel to the base with a 
distance h. It will be shown in this chapter that a maximum value of the square root of the 

determinant of the product of the manipulator Jacobian by its transpose, i.e., ^detJ^J^ , 
may be obtained from this symmetric configuration. 

85 



86 




Figure 7. 1 : A redundant 8-8 in-parallel 
manipulator 



Figure 7.2: Plan view of the redundant 8-8 
in-parallel manipulator 



First, a fixed coordinate system is placed at the center of the octagonal base as 
shown in Figure 7.2. Then the coordinates of the points Ai, A2, fii, B2, Ci, C2, D\, and D2 
on the platform are 



where 



f 



A 



fi, 



-d^ flj h 



V 



(4io 



-d, -Jj h 



42a 



- Jj h 



D, 



V2a 
^5 — z- ^5 h 



) 



B. 



D. 



' ' 2 
V 




2 ' ' 


J 


5 2 5 

V 


h 
J 


5 2 5 
V 


h 

J 



42 a a 



(7.1) 



87 



The coordinates of the points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are 



■^ 0^ 



, F, 



b \ „(b 







, F, 



2 -d. 



^ 



. G, 



fb 



d. 



f b \ 
d, - 

2 

^ ) 



\ 



, HA-d, - 



. ^D 



/ 



b 



\ 



(7.2) 



T d, 



E 



— -d. 



Same as (5.2), d^ - 



{\-2P)b 



Now the Pliicker Une coordinates of the eight leg lines of the redundant 8-8 
parallel manipulator can be obtained by counting the 2x2 determinants of the various 
arrays of the joins of the pairs of points EaAx, FaAi, ..., EdD2. From points Ea and Ai, we 

get "' / ■ ^ 



d^-d^, d.-d,, h; 



bh 



, d.h, d. 



where 



d^=—(2ap-2a-2p + \)ab and d.^ "" ^ 
4 2 



Similarly, from points Fa and Ai, 



= ^5-^4. 



bh 



From points Fb and B\, 



S,= 



ds-di, h; -^, -d^h, -d^ 

it 



bh 



d\-d^, d^-ds, h; -d^h, — — , d^ 



From points Gb and Bi, 



(7.3) 



(7.4) 



(7.5) 



^4 = 



bh 
d\-d^, d^-d^, h; d^h, - — , -d^ 



(7.6) 



From points Gc and C\, 



■^5 = 



88 



hh 

d^-d^, d^-d^, h; —, -d^h, d^^ 



(7.7) 



From points He and C2, 



V 5^ = 



hh 



d^-d^, d^-d^, h; —, d^h, -J^ 



(7.8) 



From points Ho and Di, 



S,= 



bh 



d^-d^, d^-d^, h; d^h, — , d^ 



(7.9) 



From points Eu and D2, 



5,= 



d^-d^, d^-d^, h; -d^h. 



bh 



(7.10) 



The above coordinates are not normalized and each leg line needs to be reduced to unit 
length. Then, the normalized Jacobian matrix of the eight leg lines can be expressed in 
the form 



J = 





si 


si 


Si 


Si 


si 


si 


Si 


l^i 


I2 


h 


K 


h 


k 


h 


h 



(7.11) 



where UJi, ...,k are leg lengths. 

Here, the device is in a symmetrical position so that the normalization divisor is 
the same for each leg, namely /, = /2 = . . . = /g = /, and for every leg 

[77 ~ ; (7.12) 

= J-[(2a' -2a + l)a' +V2(2a'yff-l)aZ7 + (2y9' -2y5 + l)^' +2/i'J. 



From (7.3) to (7.10), (7.1 1) becomes 



89 



-7 



d^—d^ d^—d^ d^-d^ d^-d^ d^—d^ d^—d^ d^—d^ d^—d^ 
d^ - <i, d^— d^ d^ —d^ d^— d^ <i, —d^ d, - d^ d^ —d^ d^- d^ 



bh 

2 


bh 

2 


-d,h 


d,h 


bh 

2 


bh 

2 


d,h 


-d,h 


bh 

2 


bh 

2 


-d,h 


d,h 



-d. 



-d. 



dji 

bh 

2 

d. 



-dji 

bh 

2 

-d. 



(7.13) 



Using equation (7.13), the determinant of the product JJ^ can be expressed in the 



form 



detjr=-l 



d,+d^ 











d,h 








d,+d. 





-d,h 














%h^ 












d^h 












2m.2 



-d^h {b'+4d;)h 













Qj'+Ad^W 







Ml 



(7.14) 



where 



d,^2{2a''-2a + \)a''+^I2{2ap-\)ab, 
d^ = 2(2y9' -2/3 + \)b^ + V2(2ar^ - l)ab. 

Expanding (7.14) and using (7.12), then extracting the square root yields 

/ , „T 256y/2\2aj3-2a-2/3 + l\\'b'h' 

Vdetjj'^ = L_^- ^^— J . 

(dT+d^+4h^)^ 



(7.15) 



It is important to note that VdetJJ^ = (with a,b,h^ 0) when 



90 

■ fia,fi) = 2aJ3-2a-2fi + \ = 0. (7.16) 

The plot of equation (7.16) is illustrated in Figure 7.3. Four cases that satisfy the relation 
f{a,fi) = are also drawn in Figure 7.3. When a and ;5 satisfy the relation f{a,fi) = 0, 
a redundant 8-8 parallel manipulator has finite mobility even if all its eight actuators are 
locked. A similar relation was found for a nonredundant 6-6 parallel manipulator by Lee 
and Duffy [20] and this kind of mobility was discussed in Hunt and McAree [14]. It is 

interesting to note that the two end points of the curve f(a,fi) = 0, i.e., a = 0, y9 = - , and 
a=—,/3= 0, are two cases for which the manipulator is degenerate. 




Figure 7.3: Plot of f(a, j3) = 2afi-2a-2J3 + \ = 



91 



Now, assuming the top platform size a is given and taking the partial derivative of 
(7.15) with respect to h and b respectively and then equating to zero, we obtain 

76sS\2a/3-2a-2j3 + \fa^b^h\dT+dg-4h^) 



and 



= (7.17) 



16S^f2\2aj3-2a-2fi + lfa^b^h'id^-dg+4h^)_ 

T — A = 0. (7.18) 

(dT+d,+4h^)' 

When a, b, and h are not equal to zero and 2afi -2a-2fi + l^0, from equations (7. 17) 
and (7.18), we get 

d, +d, -4h^ = 2[(2a' -2Qr+l)a' +42{2ap-\)ab + (2p^ -2p + \)b'' -2h']=Q, (7.19) 

d, -rfg +4/i' = 2[i2a^-2a + l)a^ -{2/3^ -2j3 + l)b^ +2/i']=0 . (7.20) 

Adding above two equations gives 

2dT=2[2i2a^-2a + l)a^+S(2a/3-l)ab\=0. (7.21) 

Then solving (7.21) yields 

^ V2a(2ar^-2a + l) 

"' i-2ap ■ <'-^^' 

Substituting (7.22) into equation (7.21), then solving for h we can get two solutions, here 

we only take the positive solution (again, it should be noted that the negative solution is 

simply a reflection through the base) 

, \2afi -2a-2p + \\aJ2{2a'' -2a + \) 

h=i—!- "- — ^1^ i. (7.23) 

2{\-2al3) 

Clearly, h=Q when f{a,p) = 2aP - 2a - 2y5 + 1 = . The variation of h with respect to a 

and y5 for a = 1 is shown in Figure 7.4. 



92 




0.5 



Figure 7.4: Plot of h vs. or and y5 with a = 1 



Finally, substituting (7.22) and (7.23) into (7.15) yields 

VdetJ^Jl =(VdetJJ^)_ ^4y[2(2a' -la + l^a' 



(7.24) 



where Jm denotes the Jacobian matrix for the configuration at which VdetJJ^ has a 
maximum value. It is interesting to note that this value is a function of a only and not of 



fi. When a=0, V^et J„ J^ has a maximum value 4V2a' , and when a= - , -^detj^j]^ 



becomes minimal, 2a\ Figure 7.5 plots the variation of ^detj^j^ with respect to a for 
a= 1 and shows that VdetJ^jT decreases as or increases. 

The redundant 8-8 platform with a=- and y9=- is shown in the optimal 

8 8 

configuration in Figure 7.6 and for which, from (7.22) and (7.23), b=—42a and 

31 

85 I 195 

h ^—rza . From (7.24), the corresponding value of Vdet Jj"^ is -—a . 
248 32 



93 



-E 4 






1 - 



k V- .• 4 J..... 

^^. •. .' i J..... 



0.0 0.1 0.2 0.3 0.4 0.5 
a 

Figure 7.5: Plot of -y/detj^jj, vs. orwith a = 1 




Figure 7.6: An example of redundant 8-8 manipulator in optimal configuration 



1 . 25 



with a=-,p=-, b= — ^f2a , and h 



8 ' 8 



31 



85 

248' 



94 

7.2 Implementation 
Now, from (1.2) and (7.24), the quality index for the redundant 8-8 parallel 
manipulator is given by 

V_' '- ' Vd etlJ"^ 

'^--— 3—- (7.25) 

4V2(2ar^-2a + l)2a^ 

The variation of the quality index now is investigated for a number of simple motions of 

the top platform. Here, we simply consider the case with a=y^and the top platform side a 

= 1. The base side ^ is then determined by (7.22). 

Firstly, consider a pure translation of the platform from the initial position along 

the z-axis while remaining parallel to the base. For such movement, from (7.15) and 

(7.25), the quality index is given by 

, _ 64\2afi-2a-2J3 + \fb^h^ 

^ T- (7.26) 

I — 2 

With a=/3, a= I, and b = from (7.22), dy becomes zero and ds reduces 

l-2a 

to 

. 2(2a^-2a-i-l)(l-4Qr-i-2ar')' 
''■= SS^rip ■ P-27) 

Thus, the quality index becomes 



3 
.3 



16V2|2a^-4a-i-l| (l-2a^y(2a^ -2a + \yh^ 



[(2a' -2ar + l)(2a' -4flr + l)' +2(l-2a^fh^] 
and is plotted in Figure 7.7(a) as a function of h and a. Figure 7.7(b) presents the change 
of quality index for different values of a. The variation of quality index with a(=y3) and 



95 



h is interesting. When a{=P)=0 we obtain the 4-4 platform shown by Figure 3.2, when 
a(=/J)=— we obtain another 4-4 platform shown by Figure 7.8, when ai=fi) is the 



solution of (7.16) for which a=fi= 1 — j=, the platform is degenerate and /i = (see also 

V2 

Figure 7.3). It follows that as a increases from to 1 — ;= the value of h for /l=l 

V2 

decreases whereas when or increases from 1 — pr to — the value of /i for A= 1 increases. 

V2 2 



Each value of or designates the distance between the separation points in the top platform 
and base and is our initial design parameter. Clearly, cir(=y9)=0 is the best overall design. 

We now derive an expression for ^det Jj'^ when the platform of the redundant 
8-8 parallel manipulator is translated away from its central location parallel to the base 
with height h. Assume the center of the moving platform to move to point {x, y, h), then 
the coordinates of the points Ai, A2, Bi, B2, Ci, C2, Du and D2 on the platform become 



x-d^ y + d^- 



V2a 



h 



B,\x-d,+^ y-d, h 



x + d^ y + d^- 



Sa 



5. 



J 

x-d^-V y-^d^ h 



J 



x + d^ y-d^ + 



4ia 



x-d^ y-d^ + 



Sa .1 



(7.29) 



D, 



x + d^- 



y + d^ h 



D. 



x + d, — y-d, h 



96 




0.3 

2 p^a 





1.0 - 


::::;( 




T X^XV/'^^x 


■ 






aAA \ 


\p=a 




0.8 - 


A/ A \ \ 




'^ 


aA \ IV''^""^ ^ 




X 

o 

T3 

C 


0.6 - 


il • \ v\ U«=2^5 




/A: \ \ .3v^ 




ir 




||/\; \ 3t^\ d=i/4 




m 


0.4 - 


1/ / V V''1\ vy^ i 




..3 






o 




ll/ \ \ ^ ^X: \J/"~ 


U8 




0.2 - 


Pl^^^ 


^a=0 






1 1 1 "! ' 





3 



0.0 0.5 1.0 1.5 2.0 2.5 3.0 
Height h 



(a) (b) 

Figure 7.7: Quality index for platform vertical movement 




Figure 7.8: Reduction of the size of the redundant 8-8 in-parallel 
manipulator when a{=P) = — 



97 

The coordinates of points Ea, Fa, Fb, Gb, Gc, He, Ho, and Ep on the base can be 
found from (7.2). Thus, the Plucker line coordinates for each of the eight leg lines are 
determined as 



S,= 



^ , , . ^ . . ^^ ^ , b(x-d^)-d^(2y-^f2a + 2d,) 
x+d^-d^, y-di+d^, h; , d^h, ^ 1L_£ il 



, (7.30) 



S,= 



x-d^+d^, y-di+d^, h; -—, -d^h, ^- *-^^ ^ 

^ 2 



, (7.31) 



53 = 



x+di-d^, y+d^-d^, h\ -d^h, 



bh d^{2x + ^[2a-ld^)+b{y-d^ 



JC + 



. , J ^ J I, J I. bh d,(2x+^/2a-2d,)+b(y+d,) 
di-d^, y-d^+d^, h; d^h, -—, — ^ ^- — ^-^ ^ 



. (7.32) 



, (7.33) 



^5 = 



w ^^ ^^ ^ u ^^ ^ u b{x+d,)-d,{2y+^[2a-2d,) 
x-d^+d^, y+d^-d^, h; — , -d^h, *-^ ^— 



, (7.34) 



■^6 = 



x^d,-d„ y^d,-d„ h- ^, dA J^--d.)^d,i2y^^a-2d,) 



, (7.35) 



5,= 



^A ^A A .A u A I. bh d^{2x-42a + 2d,)+b{y+d.) 
x+d^+d^, y-d^+d^, h; d.h, — , — ^ ^- ^^ ^— 

2 2 



, (7.36) 



A ^A .A A I A 1 bh d^{2x-42a + 2d.)-b{y-d.) 
x-di+d^, y+d^-ds, h; -d^h, —, -^ —^ — ^-^ 5Z 

2 2 



, (7.37) 



where du d^, and ds can be found from (3.13), (5.2), and (7.1) respectively. The above 
coordinates are not normalized and each row must be divided by the corresponding leg 
length. 

The Jacobian matrix can then be constructed by using (7.11). Further, ^detJJ^ 



becomes 



98 

II 1 1 1 1 1 1 
where the leg lengths are 

l,=^{x + d,-dj +{y-d^+d,y +h\ l^=^{x-d,+d,f +{y-d,+dj +h\ 

l^ =^{x + d,-dj +{y + d,-d,y +h\ l^=^{x + d,-dj+{y-d,+dj+h\ 

l,=^{x-d,+dj +{y + d,-dj +h\ l^=^{x + d,-d,f +{y + d,-d,f +h\ 

l^ =^{x-d,+d,f +{y-d,+dj +h\ l^=^{x-d,+dj+{y + d,-d,y+h\ 

Substituting (7.38) into (7.25) with a=/3, a=l, Z? = ^^^^^' ~^f "^^^ from (7.22), and 

l-2a 

2a^-4a + l 



h = h_ = 



■^2(2a^-2a + l) 

from (7.23), the quality index for the platform 



2(l-2ar0 
translation of the optimum redundant 8-8 parallel manipulator will be determined. 

Figure 7.9(a)-(d) illustrate the quality index surfaces and contours for various 
values of a as the platform is translated away from the central location while remaining 
parallel to the base. When a(=fi) = 0, the manipulator reduces to the redundant 4-4 
parallel manipulator whose quality index surface and contours have been shown in Figure 
3.5. Comparing Figures 7.9(a)-(d) with Figure 3.5, it is clear that cases (b), (c), and (d) 
are unacceptable designs. 



99 




1.0 - 



0.5 



P^O.O 



-0.5 - •• 



■1.0 




1 -1 



-1^ -0.5 0.0 0.5 1.0 
X 



(a) «(=;?)= 1 at ;i„=l 




1.0 -■■■. 



0.5 - 



?s0.0 



-0.5 



-1.0 - 




1 -1 



-1.0 -0.5 0.0 0.5 



1.0 



(b)a(=;^=|at/.„=^^0.21 



100 




1.0 



0.5 



^0.0 



-0.5 



-1.0 



(c)a{=/J) = - at /i„= — = 0.08 
4 ""28 



»)\- 



—I 1 1 1 1— 

-1.0 -0.5 0.0 0.5 1.0 

X 




1.0 



0.5 



?^0.0 - 



-0.5 - 



-1.0 - 




1 -1 



1 1 1 1 r 

-1.0 -0.5 0.0 0.5 1.0 



1 8S 



Figure 7.9: Quality index for platform horizontal translation 
with different values of a (= y9) 



101 



Figure 7.10 illustrates the side view of the moving platform rotated 6^ about the 
^''-axis, which is drawn through the center of the top platform and parallel to the j-axis 
located in the fixed base. The center of the platform is located at height h. Then the 
coordinates of the vertices A i, A2, Bi, B2, Ci, C2, £>i, and D2 become 



f 



-Jt. 

-d.cosd^ \-d. h-d. sind,. 



( V2a 
A2 d^cosO^ T- + d^ h + d^sinO,, 



B, 



(—-d,)cose^ -d, h + i—-d,)sme^. 



B. 



C, 



I>, 



( 

X 

V 

( 

1 

V 

( 
\ 

( 

V 
( 



{^-d,)co%e^ d, h + i^-d,)smd^ 



4i 

d^ cos 6^ d^ h + d^ sin d^ 



-d^cos$y 



J 



,4la 



42a 



\ 



-( rf,)cos^,, d, h-C 

2 ' - ' 2 



-J5)sin^^. 



D. 



Ma 



,Sa 



-(— — d^)cosey -d^ h-{— rf5)sin^^ 



(7.39) 



Applying Grassmann's method, the Plucker coordinates for the corresponding 
lines $1, $2, ..., $8 are given by 



S,= 



d^-d^cosO , -di+d^, h-d^sinO; 



2 



--b(h-d^smd^), d^(h-d^smOy), 



J4 {^|2a - 2^5 ) - bd. cos 6^ 



(7.40) 



102 




MGb) 



Figure 7.10: Platform rotations about the j '-axis 



S,= 



-d^+d^cosOy, -di+d^, h + d^sinO^; 



^u^u. J ■ n ^ J fu, J ■ a ^ d^(yf2a-2ds)-bd^cose^ 



(7.41) 



53 = 



b - {■yjla - Id. ) cos 9 



-, d.-d., h + 



(V2a -2^/5) sin 0^, 



^ ^, (V2a-2rf5)sin^,^ b ^, {42a -Id,) sin 6^ 
-d,(h+ -), --(h+ '-), 

bd, - c?4 ( V2a - 2d, ) cos 0^ 



(7.42) 



103 



^4 = 



b-{42a-2d^)cose^ {■sfla - 2d ^) sin d 
-, -d^+d^, h-^ 



y . 



(V2a-2rf5)sin^ ^ (V2fl - 2^, ) sin ^„ 
dAh+ '-), --{h+ :^ '-), 

bd^ - d^ {s2a - 2d^ ) cos 6^. 



(7.43) 



•^5 = 



-d^+d^cos6y, d^-d^, h + d^sindy', 

d^ ( V2a - 2d. ) - bd. cos ^ 



-b^h + d^sind ), -d^ih + d^sind ), "'* 



.^j •'"5^^'^^y 



(7.44) 



Se = 



1 



d^-d^cosd , d^-d^, h-d^s'm6; 



-bih-d^sind ), d^ih-d^sind ), - 



d^ (V2a -2d,)- bd, cos 9 



(7.45) 



S,= 



b-{y[2a-2d,)co%e^ {yl2a-2d,)sm0^ 
'-, -d,+d„ h ^^ ^ 



{42a-2d,)sme^^ b , {42a-2d,)smd, 
2 



. (c IMIIC,, 



bd^ - d^ ( V2a - 2d. ) cos 6 



SJ^^^^y 



S.= 



b-{^l2a-2d^)cosdy 



, d^-d^, h- 



(\2a-2d.)sm0 



(V2a-2rf5)sin^ ^ (^/2a-2d,)sm0^ 
-d,ih ^ ), -ih ^ -). 

bd^-dA42a-2d^)co&e^ 



(7.46) 



(7.47) 



104 



Since the configuration of the manipulator keeps symmetric about the ;c-axis 
during the platform rotation about the j '-axis, from Figure 7. 10, we have 



/, =l^=^(d^cos9^.-dJ'^+(d^ -d^f+(d^smd^. -hf, 
I2 ^h = -J(^5 cos^^ - ^4 )^ + (^1 -d^f+ (flfj sin 6^ + hf , 



'3 ~ M ~ ■ 



h ~ '■» ~ ■ 



{y!la-2d^)co^d^-b 



+ {d,-d,f + 



\[2a-2d 



V 



-sin^„ +h 



(7.48) 



V 



{42a-2d,)cosd^-b^ ^^ ^. (42a-2d, . 

+ (^4-^5) + 



^-sind-h 

2 ' 



J 



The corresponding Jacobian matrix J then can be obtained by (7.11). Substituting 
-y/det JJ^ into (7.25), the quality index for such platform rotation will be determined and 
it is too large to be expressed here. 

Figure 7.11(a) plots the quality index surface with respect to 6^ and fi for the 

redundant 8-8 in-parallel manipulator with a=p, a=l, h = — ^ — j— ^ -, and 



h = h = 



2a -4a -1-1 



^2{2a^-2a + l) 



2(1 -2a^) 



\-2a' 



. The change of the quality index for several 



different values of y0 is shown in Figure 7. 1 1(b). 

It should be noted that since the redundant 8-8 in-parallel manipulator is fully 
symmetric, the same result can be obtained for the platform rotations about the ;c'-axis, 
which is drawn through the center of the top platform and parallel to the jc-axis located in 
the fixed base. 



105 




-90 -60 -30 30 60 90 
Rotation Angle 9 (degree) 

(b) 

Figure 7.1 1: Quality index for platform rotations about the x'- and y '-axes 



Figure 7.12 illustrates a plan view of the moving platform rotated Ot, about the 
vertical z-axis through its center, the legs being adjusted in length to keep the platform 
parallel to the base at a distance h. The x and y coordinates of the vertices A], Ai, B\, B2, 
Ci, C2, Di, and D2 are given by 



x^^ =rsin(d^ +</)), 
Xb, =rcos(0^-(p), 
XB^=rcos{d^+(p), 
Xq =-rsin{e^-<f>), 
Xc, =-rsm(0^+<p), 
Xi^ =-rcosid^-<p), 
Xd, =-rcos(e^+<p). 



>'a, =-rcos(e^-(p), 
Va, =-rcos(0^+<p), 
yg^ =rsm(0^-(p), 
ya, =rsm{e^ +(/)), 
^c, =rcos(6>^-0), 

y^ =-rsin(^^-^), 
Jd, =-rsm(0^+<p). 



(7.49) 



where r-—yl2(2a^ -2a + l) and ^ = arcsin 



a 



yl{2a^-2a + \) 



106 




Figure 7.12: Platform rotations about the z-axis 



The complete set of coordinates of points Ai, Az, B\, B2, C\, C2, D\, and D2 are 
therefore 
Ak, A h), B^[xg^ y,^ h), C^x^^ y. h), D,(x^ y h\ 



^k, yA, h\ bJ^x 



>'fi. 



h\ QL j. h), dXxd, Jo, h). 



(7.50) 



The coordinates for the corresponding lines $1, $2, ..., $8 are then given by 



S,= 



b , bh _, , bx. 



(7.51) 



^2 = 



b , bh , , bx. 

XA,-d„ >'^,+-, h; -—, -d,h, -^ + d,y^^ 



(7.52) 



h hh by 

^B,--. yB,+d„ h; -dji, -—, d,x,^+^ 



(7.53) 



107 



^4 = 






(7.54) 



■^5 = 



bh 



bXr 



Xc,-d„ >'c,-^. h; y, -d,h, ^ + ^4>'c, 



(7.55) 



Se = 



Xc,+d„ yc,--' h; —, d,h. 



bXr 



-d,yc. 



(7.56) 



•^7 = 






(7.57) 



S,= 



^D^ + j' ^02+^4. ^; -^4^. 



2 



C/^Xn - 



^^D, 



' '^4-^D- 



(7.58) 



From Figure 7.12, we have 1^=1-^=1^=1^ and l2-h-h-K- The 



corresponding Jacobian matrix can now be determined by (7.11), and further ^detJJ^ 
becomes 



VdetJJ^ =■ 



A^!2a'b^h\2pa-2a-2p + Xf cosd^ 



Pi' 



(7.59) 



where 



/, = j(r sin(^^ -(/>) + dJ+(r cos(^^ - <^) - 



^"l 



+ /^\ 



■ l,=J{rsm(0^+<P)-dJ + 



( 



b\ 



rcos{6^ +(p) — 



+ h' 



Substituting (7.59) into (7.25) with a=fi, a=l, b = ^^f^^ — ^^, and 

l-2a' 



h = h = 



2a^ -4a + l 



■^2{2a^-2a + l) 



2(1 -2a') 



, the quality index will be determined as a 



108 

function of 6^ and a. This is plotted in Figure 7.13(a). The variation curves of the quality 
index for several different values of a are shown in Figure 7.13(b). 




-90 -60 -30 30 60 90 
Rotation Angle 9 (degree) 



(a) (b) 

Figure 7.13: Quality index for platform rotation about the z-axis 



From Figure 7.13(a) and (b), we can see the manipulator has the highest quality 
index ;i = 1 when ^ = 0°, and A = (singularity) when 61, = ± 90°. Again, from Figures 
7.11 and 7.13 we can see clearly that better designs are obtained as a {=P) reduces to 
zero. Hence the best 8-8 parallel manipulator design is obtained when the pair of 
separated joints in the base and top platform are as close as possible. 

Since from (7.59), -y/detjj"^ =0 when (9z = ±90°, a redundant 8-8 parallel 
manipulator always become singular when its platform rotates 6*2= ±90° about z-axis 
from its central symmetric position. Figure 7.14 illustrates the singularity position for 
^, = 90°. 



109 




Figure 7.14: Plan view of the singularity position of redundant 8-8 in-parallel 

manipulator when dz = 90° 



Similar to the redundant 4-4 and 4-8 cases, when ^=±90°, the moving platform 
of the redundant 8-8 parallel manipulator can move instantaneously on a screw reciprocal 

to the eight leg forces on the z-axis with pitch h^ =H ^^. This is because 

when ^z=±90°, from (7.51) through (7.58), the component of moments about the z-axis 

V2 
for each of the eight legs all are equal to ± — ab{\ - lap) . 

4 



CHAPTER 8 

THE FORWARD KINEMATIC ANALYSIS OF THE SPATIAL REDUNDANT 

8-8 IN-PARALLEL MANIPULATOR 

In this chapter, the forward kinematic analysis for the redundant 8-8 parallel 

manipulator shown in Figure 7.1 is studied. Since there are no double-spherical joints in 

the 8-8 manipulator, the method used for solving the forward kinematics of redundant 4-4 

and 4-8 parallel manipulators does not work for the 8-8 case. We use a different method 

to determine the location of the moving platform given the actuator displacements of all 

the eight legs. 



. 8. 1 Coordinate Systems 

First, two coordinate systems, 1 and 2, have been attached to the fixed base and 
the moving platform respectively as shown in Figure 8.1. The origins of the coordinate 
systems 1 and 2 are located at points E and A receptively. The coordinates of points Ea, 
Fa, Fb, Gb, Gc, He, Ho, and Ed on the base are known in terms of the coordinate system 1 
and 



>" 




'{\-P)b 




b 




>- 





'p - 

*^Fa 





'P - 


il-fi)b 


'P - 


b 





















Ed 



"0" 




' b' 


^ 


'p = 

*^Fb - 


Pb 











Gc 



'{\-P)b 







b 


'P - 


{\-p)b 











(8.1) 



110 



Ill 



while the coordinates of points A\, Aj, B\, Bi, C\, C2, D\, and D2 on the platform are 
known in terms of the coordinate system 2 and 



P. = 








(1- 


a)a 




a 




aa 




aa 


^P - 





^P - 


{\-a)a 


^P - 


a 


? 





' 't 

















aa 




a 






'{\-a)a 










' % = 


aa 


> 


2p _ 


a 


^P - 


(l-a)a 























(8.2) 




Figure 8. 1 : Coordinate systems of a redundant 8-8 in-parallel manipulator 



112 

The objective of the forward analysis of the redundant 8-8 parallel manipulator is 
to determine the coordinates of points Ai, A2, 5i, B2, Ci, C2, Di, and D2 measured in the 
coordinate system 1 given eight leg lengths /, (/= 1, 2, .... 8). Thus, we need to relate the 
position and orientation of the coordinate system 2 in three-dimensional space to the 
coordinate system 1. Once this has been accomplished, it is possible to transform the 
coordinates of any point in coordinate system 2 to coordinate system 1 . 

Here, we let the nine coordinates of three non-coUinear points from the moving 
platform be chosen to represent the position and orientation of the coordinate system 2 in 
space. Then, the coordinates of the other points on the platform are expressed in terms of 
these nine coordinates. 

Assume the coordinates of points A, B\, and D2 measured in coordinate system 1 
are 



Pa = 



"^a' 




"^b" 




'^d' 


^A 


'P = 


^B 


'P = 


)'d 


.^A. 




.^B. 




.^D. 



(8.3) 



Since vector V^- P^ is along the j:-axis of coordinate system 2 while vector 'Pp -'P^ 
is along the j-axis, the z-axis will be along ('Pb_ -'Pa) xC'Pd^-'P^) . Therefore, the three 

unit vectors along the coordinate axes of the coordinate system 2 measured in the 
coordinate system 1 are 



'x =— ^ 



{\-a)a 



(8.4) 



'P -'P 

J2 ~ 



(1 - a)a 



(8.5) 



113 



('Pb-'PJx('Pd-'PJ 



They specify the orientation of the coordinate system 2 relative to 1 and can be 
represented as a 3x3 matrix 

lR = ['x, 'y, 'zj. (8.7) 

Since point A is the origin of coordinate system 2, the coordinates of points Ai, A2, 
^2, Ci, C2, and Di may now be expressed in the coordinate system 1 as 

'P,,= 'P.+ lR^, 

Pflj = Pa + 2^ Pbj » 
Pc, = Pa + 2^ Pc, ' 
Pc2 = Pa + 2^ Pc2 ' 

Pd, = Pa + 2^ Pd, ' 

where ^P^^, ^P^^, ^P^, ^P^^, ^P^, and ^P^^ are the coordinates of points A 1,^2, ^2, Ci, 
C2, and D\ measured in the coordinate system 2 and given by (8.2). 

The forward problem reduces to determine the nine coordinates of points A, Bi, 
and D2 in the coordinate system 1 such that the moving platform satisfies the distance 
constraints with the base. 

8.2 Constraint Equations 
Since the eight leg lengths, /, (i=l, 2, ..., 8), have been given for the forward 
analysis, the location of the moving platform must comply with these constraints. Thus, 



114 

the geometry constraint equations between the moving platform and the fixed base can be 
written in the coordinate system 1 as 

('p,,-'p,J•('p,,-'p,J=/^ : (8.9) 

Cp^^-%)-CK,-%J = il (8.10) 

']-, ('Pb.-'Pf,)-('Pb,-'I\) = /3, (8.11) 

,, . (%^-'PaJ-CPB,-'Pc,) = ll (8.12) 

('Pc,-'Po,)-('Pc,-'Pg,) = /5'. (8.13) 

('Pc^-'P„^)-('P,^-'Ph^) = /^ (8.14) 

('Pd,-'PhJ-('Pd,-'Ph,) = /7, (8.15) 

('Pd,-'Pe.)-('Pd,-'PeJ = ^8- (8.16) 

There are another three constraint equations representing the distance constraints 

on points A, B\, and Di on the platform, which have been used to define the position and 

orientation of the coordinate system 2 relative to 1. These equations are written in the 

coordinate system 1 as 

("PB,-'PJ('PB.-'PJ = (l-a)V, (8.17) 

(•Po^-'PJ-('P,^-'PJ = (l-«)V, (8.18) 

('Pd,-'Pb,)('Pd,-'Pb,) = 2(1-«)V. (8.19) 
Thus, we have 1 1 equations in 9 unknowns {x^, y^, Zk, x^, y^, zq, x^, yn, and zd)- 

8.3 Equation Solution 
Substituting (8.1), (8.3), (8.8) into equations (8.11), (8.10), and (8.16), then 
expanding and regrouping yield 



115 

zl=-xl-yl+2(x^x^ + y^y^ + Zj,Zo)-2J^y^+{\-2a + aV+fi^b'-l^, (8.20) 

Zb = -4 -yl+ 2bx^ + Ipby^ -(I + 13' )b' + 1] , (8.21) 

zl = -xl -yl+ 2fiby^ - fi'b' + /^ (8.22) 

Further, substituting (8.1), (8.3), (8.8) and the above expressions for zl, zl, and zl into 
equations (8.9), (8. 14), and (8. 19) yields 

ZAZB=-XAX^-y/,ys + l3b(x^-x^) + b(2-a-fi + a/3)x^ + 

(8.23) 
biy^-yo+J3ya)-bil-a-2/3 + a/i)y^+q^, 

ZbZd = -^B^D - >'b>'d + bx^ + fib(yB + yo) + ^2 ' (8.24) 

o, /h[axn+(l-3a)yr,] 

ZaZo = -^A^D - >'a>'d + fibx^ + ^ ; -^ + ^3 . (8.25) 

l-2a 

where qi, qj, and q^ are known constants and are defined in Appendix B. 

Now substituting the expression for zaZd in (8.25) into (8.20), the expression for 

z\ becomes . - >. , - . - « 

zl = -xl -yl+ 2pbx, + '^'^^^y^' ^"^ + 2^3 + (1 - 2a + aV + P'b' - ll . (8.26) 

Five equations that are linear in xa, yA, xb, ys, xd, and >'d are generated by 
substituting (8.1), (8.3), (8.8), (8.21) - (8.26) into equations (8.10), (8.12), (8.13), (8.15), 
and (8.17): 

^l^A +^2)'a +^-3^b +^4)'b +^5^D +^6>'d + ^7 =0 = 12,3,4,5). (8.27) 

Solving the above five linear equations with xa, yA, ye, xd, and >'d as unknowns, we 
obtain 



116 

Xq — —Xq + q^ 
1 

>'a=— ; -^8+96 

^"^ (8.28) 

a 
l-a 

where the constants are defined in Appendix B. 

Now, there is only one unknown xq need to be determined. To find the solution 

for ;cb, we need to construct more equations. Observing the left hand sides of (8.21) - 

(8.26), we may formulate the following six identities 

Eq2 =(zI)(Zj^Zd)-(z^Zs)(ZsZo) = 0, 

Eq^ = (ZdXZa^b) - (Za^dXZbZd) = 0. 

(8.29) 
Eq,=izl)izl)-iz^z^)'^0, 

Eq,={zl){zl)-iz,z^)'=0, 

Eq,=(zl)izl)-iz^Z^)'=0. 

Substituting (8.21) - (8.26), and (8.28) into the above identities, we finally get six 
equations in only one unknown, ;cb: 

Eq,{x^) = f^Nyxi=0 (/ = 1,2,...,6) (8.30) 

where the constants are defined in Appendix B. 

Choosing any 5 from the above 6 equations, we can determine the value of jcb. 
Here, we select the first 5 equations and they can be represented in matrix form as 



117 



Ny=0 



(8.31) 



where 



N = 



^14 A^,3 A^,2 A^n A^,o 



N N N N N 

JY24 iV23 ^^22 ^'21 ^''20 



^34 A^33 A^32 A^3. ^^30 



A^44 ^43 A^42 A^4I A^40 



^54 A^53 A^52 ^5. ^^50 



y = 



1 



The matrix equation (8.31) may be thought of as a homogeneous linear system in five 
unknowns. The necessary and sufficient condition under which these linear equations 
have non-trivial solutions is that the determinant of the coefficient matrix N must vanish. 
The five equations represented by (8.31) may now be rearranged into the form 

Wx = r (8.32) 

where 



W 



N,, A^,3 yv,2 yv„ 



"24 ^23 ^22 '*21 



N^ A^33 ^32 A^31 



A^44 A^43 A^42 A^4, 



^54 ^53 A^52 A^5. 



x = 



r = 



-A^, 



10 



A^, 



20 



-N 



30 



-A^ 



40 



-N 



50 



Equation (8.32) represents five linear equations in four unknowns. The vector x may be 
solved for by selecting any four of those equations. The term xq is the fourth component 
of the vector x and unique value for this term is thereby determined. Corresponding 
values of xa, ^a, ^b, xd, and ^d can be obtained from (8.28). Finally, z\, Zb, and zd can be 
determined from (8.22). It should be noted that there are two opposite sign solutions for 



118 

Za, Zb, and zd, respectively. However, they cannot be combined arbitrarily since they 
must also comply with the three equations in (8.25). Therefore, only two groups of the 
solution of za, Zb, and zd are possible and the manipulator has two configurations for a set 
of given leg lengths. 

8.4 Numerical Verification 
In this section, a numerical example is presented for a redundant 8-8 parallel 
manipulator to verify the analysis. The dimensions of the manipulator are measured in an 
arbitrary length unit and given as follows: 

platform side a = 10, base side ft = 15, 

A set of leg lengths are given as 

/, =13.29955, /^ =14.24887, /, =9.77545, /^ =11.25375, 
/, = 11.60143, /g = 15.41449, /7= 15.63800, /8 = 18.01 133. 

The numerical results are presented in Table 8.1. A unique configuration may be easily 
determined by checking the sign of z coordinate of one of the platform joints. It has been 
verified by an inverse kinematic analysis that all solutions reproduced the correct leg 
lengths. 

Same as the redundant 4-4 and 4-8 parallel manipulators, the above solution failed 
when the platform rotates about the z-axis (Figure 7.12). At such special configuration, 
the odd leg lengths are all equal and /, = /, = /j = /^ = / . Further, the even leg lengths are 
all equal and 1^=1^=1^=1^=1'. Therefore, equation (8.9) through (8.16) become 
linearly dependent. 



119 



Table 8.1: Numerical results of the redundant 8-8 in-parallel manipulator 



No. 


'P 


'P 


•Pb, 


'^ 


'Pc. 


'P 


'P 


'P 


1 


" 4.667" 

3.644 

12.482 


r 5.717] 

2.987 
[11.799] 


["12.857] 

5.191 
[ 8.943 J 


[13.592" 

6.398 

|_ 8.912_ 


[12.333] 

13.856 

[1 1.518] 


[11.283" 

14.513 

[l2.201_ 


[4.143" 

12.309 

[15.057 


[ 3.408] 

11.102 

|_15.088j 


2 


4.667' 
3.644 
-12.482 


r 5.717] 

2.987 
[-II.799J 


[12.857] 

5.191 
[-8.943J 


[13.592" 

6.398 

[-8.912_ 


[ 12.333] 

13.856 

[-11.518J 


[11.283" 
14.513 
[-12.201 


[ 4.143" 

12.309 

|_-15.057_ 


[ 3.408] 
11.102 

|_-15.088j 



Table 8.2: A numerical example for the special case of 
the redundant 8-8 in-parallel manipulator 



No. 


'Pa, 


'P 


'Pb, 


'Pb. 


'P 


'P 


P 


P 




"4.135" 


[2.961" 


[-4.889" 


[-5.677" 


[-4.135' 


[-2.961' 


[ 4.889" 


[ 5.677] 


1 


4.889 


5.677 


4.135 


2.961 


-4.889 


-5.677 


-4.135 


-2.961 




8.675 


[8.675_ 


[ 8.675 


[ 8.675 _ 


[ 8.675_ 


[ 8.675 


[ 8.675 _ 


[ 8.675] 




" 2.961" 


[4.135" 


[ 5.677" 


[ 4.889" 


[-2.961" 


[-4.135" 


[-5.677" 


[-4.889] 


2 


-5.677 


-4.889 


2.961 


4.135 


5.677 


4.889 


-2.961 


-4.135 




15.715 


[l5.7i5_ 


[15.715J 


[15.715J 


[l5.715_ 


[l5.715_ 


[l5.715_ 


[15.715J 




"4.135] 


[ 2.961" 


[-4.889] 


[-5.677] 


[-4.135] 


[-2.961] 


[ 4.889] 


[ 5.677] 


3 


4.889 


5.677 


4.135 


2.961 


-4.889 


-5.677 


-4.135 


-2.961 




-8.675] 


[-8.675J 


[-8.675] 


[-8.675J 


[-8.675J 


[-8.675J 


[-8.675J 


[-8.675] 




2.961] 


[ 4.135] 


[ 5.677] 


[ 4.889] 


[ -2.961] 


[ -4.135] 


[ -5.677] 


[ -4.889] 


4 


-5.677 


-4.889 


2.961 


4.135 


5.677 


4.889 


-2.961 


-4.135 




-15.715] 


[-I5.715J 


[-I5.715J 


-15.715] 


-15.715] 


[-I5.715J 


[-I5.715J 


[-I5.715J 



120 

For example, when the leg lengths become 

/, =/3 =/j =/^ =/ = 18 and l^=l^=l^=l^=r=l6 

for the same redundant 8-8 parallel manipulator as the first example, two configurations 
above the base plane with another two reflected through the base are obtained as shown 
in Table 8.2. 



CHAPTER 9 
CONCLUSIONS 

Redundant in-parallel manipulators present many interests in various applications: 
increase dexterity, reduce or even eliminate singularities, increase reliability, simplify the 
forward kinematics, and improve load distribution in actuators. This work has studied 
several spatial redundant parallel manipulators. 

The quality index for redundant parallel manipulators has been defined as a 
dimensionless ratio which takes a maximum value of 1 at a central symmetrical 
configuration that is shown to correspond to the maximum value of the square root of the 
determinant of the product of the manipulator Jacobian by its transpose. A quality index 
has two clear meanings so far. When A=0, a platform is in singular condition and when 
A = 1, it is in its optimal geometry static configuration. However, when A is neither zero 
nor one, it is hard to say exactly how much one configuration is better than another. One 
can not say that a configuration with A=0.8 is twice as good as a configuration with 
Ji=OA without further analyses. However, a quality index helps in the design platforms 
by setting dimensions that give best quality index value. Also, it gives an idea of certain 
designs that must be prevented as they would lead to zero or lower quality indexes. The 
quality index reflects singularities, and therefore gives an indication of the safe regions 
within which the manipulator can be maneuvered and controlled. 

Using quality index, variable motions are investigated for which a moving 
platform rotates about a central axis or moves parallel to the base. It shows that the wider 



121 



122 

the range of high quaUty index, the better the design of a parallel manipulator. Thus, the 
quality index can be used as a constructive measure not only of an acceptable operating 
workspace but also of acceptable and optimum design proportions. 

Additionally, the redundant 4-4 parallel manipulator contains double-spherical 
joints. There are eight of them and they are the source of critical practical difficulties 
since they can produce serious mechanical interference. There appears to be no 
reasonable alternative than to accept a reduction in the maximum quality index through 
separation by fairly short distances of some or all of the double-spherical joints. It 
appears at this time that it may be best to separate the double-spherical joints as per the 
arrangement of the 8-8 platform shown in Figure 7. 1 in order to avoid the mechanical 
interference problem while keeping the reduction of quality index in an acceptable range. 

The forward kinematic analysis of redundant parallel manipulators has been 
performed for which it is required to determine the position and orientation of the 
platform given the leg lengths. 

The scope of future work includes the development of strategies for redundant 
parallel manipulators for various applications, a comparison of the performance of 
nonredundant and redundant parallel manipulators and an assessment of the advantages 
and costs of redundancy. 



APPENDIX A 

CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE 

REDUNDANT 4-4 IN-PARALLEL MANIPULATOR 



A:, = Ibl^s. 



k^ =2abs^s^ 



k,=b'-2abc.c.+ll-ll+ll-l^ 



^4 ~ ^^'■2^^ 



k,=a^- labc^ c^ + 1] - ll + ll - 1] 



k^ =2abl^s^s^ 



k^ =2a^bc^s^sl 



-2i „2 



ks=ab c^s^^ -2a bc^c^^s^^ + ^kc^s^, '^^^0,^ +"^6^^,^ '^^^^^ + 



2a^bc^/^s^^-2abl^sls^^ 

k,,=-2ab\s^ +a'bc^^s^^ +bl^c^^s^ -bl^c^^s^ + bl'.c^^s^ -bl^c^^s^ + 
^12 =«X^^ -2«^\*^ +b%s^ +2abl,c^c^s^ -bl^c^s^ -bl^c^s^ + 



123 



124 



M,3 = 2aWs^sl(2abc^c^ -labclc^ -a^s^ -b^sl +6abc^c^sl -Ibc^y^ - 
2ac^l,sl + 2l',sl + llsl + llsl - 2/^J + l^s^ + l^s^ ) 

M,2 = absl(2a'bc^c^ -2a'bclc^ +4a^bc^l^ -Aa^bcll^-2abc^c^ll+2abclc^ll - 
2abc^c^ll + 2abclc^ll + 2abc^c^ll - 2abclc^ll - 2abc^c^ll + 2abclc^ll - 
2\a'b^sl ^Aa'bc^c^sl +6ab'c^c^sl +4a^bc^l^sl^ +Sab\l^sl -Sa^ljsl - 
Sb^l^sl - 6abc^ c^ llsl + 2bc^ l^sl + 2ac^ I'.sl + 1^^ + a^l^sl - 6abc^ c^ lls^ + 
2ac^lf,sl -lUlsl +bHlsl -Aabc^c^llsl +2bc^l,l^sl +2ac^lf,sl -lUls^ - 
lUlsl -aHlsl -b'llsl ^lOabc^c^llsl -Abc^l.llsl -4ac^l,iy^ ^AlUlsl + 

72/2„2 . /2/2 2 _/4 2 + ^2,2 2 _ g -,/,^ ^ /2 2 , 2bc I P <!^ + 2aC I l^ <i^ -l^l'^<!^ - 

lll'sl +lll',sl ^b'llsl -Aabc^c^llsl +2bc^l,llsl -l^sl -l^sl ^lUlsl - 
iXsl +24a'bX -nab\l,sl -4a'bc^l,sl +4a'bcll,sl +2AaWslsl - 
Sa'bc^l,slsl-24a'bXsl) 

Mu = -s^s^(4a'b'+4a'b'-2a'b\c^ -"^^'^'cl^^ -4«'^'c^< -a'bc^l, + 
4a^b'^c^l^ - ab*c^l^ + 4a^b'^cll^ - 5a^bH\ - 2a'bc^c^l\ - 2ab^c^c^l\ + 
Sa^bc^ll + 2>ab^c^ll - 2abc^c^l^ - 4a^b^l^ + 2a^bc^c^ll + 2a^bc^l^ll + 
ab^c^lj^ -4aWl^ +2ab\^c^l^ +3a^bc^lJ^ +ab^c^l^l^ -a^l^l^ - 
2abc^c^llll + ac^lHl - 2abc^c^l^J^ + ac^lf^l] +SaWll - 2a'bc^c^l^ - 
2ab\c^ll -4a'bc^l,ll -4ab\lj', +a'l',l', +b'l',ll + 4abc^c^l',l', -bc^lHl - 
ac^llll^2abc^c^llll -ac^l.lH^ +2abc^c^l^l', -bc^lj'.l', -ac^l.l^l', +l',l^l', - 
2abc^c^lt + bc^l^ll + ac^l^ll - l^l^ - 4aWl^ + 2a'bc^c^l^ + a^bc^lj^ + 
3ab\l,l^ -bHll^ -2abc^c^lll', +bc^l'j^ -2abc^c^l^l^ +bc^ylllj + 
ac^yie, -lllll^ + 2abc^c^lie, -bc^lfj^ -ac^ljH^ +l',l'j^ -4aWl^ + 
2ab\c^ll +a'bc^l,ll+2ab'c^l,ll-2abc^c^llll+2abc^c^llll -bc^l.l'ji - 



125 



\2aVc^l,sl -Aa'b^cll^sl +4a^b^iy^ +6a^b^llsl +6a'b^l^,sl - 
4ab\l,l^sl -Ma'bHlsl +Sab\l,iy^ +6aWl^sl -4ab\l,l^sl + 
eaWllsl -Aab'cJJlsl -6a'b'sl -ba'b'sl +Sa'b'c^c^sl +4a'b'clc^sl - 






.3. 2„ , 2 



.2i.2;2_2 



'01 ^ ^ 



"^ ^ ^ 



4a'b\l,s'^ -Sa'b\l,s'^ +4a'b%s'^ +6aWllsl -4a'bc^l,llsl + 



ea'b'llsl -4a'bc^l,llsl -\2aWiy^ +Sa'bc^l,iy^ +6aWiy^ - 
4a'bc^l,l^sl +6aWllsl +6a'b'slsl +6a'b*slsl -4a'b\c^slsl + 
\2a'b\l,slsl +Sa'b\l,slsl -4a'bY,slsl -6a'bY,slsl -6a'b'l^slsl + 



M,,=5a'b'sl -2a'b\c^sl -2a%\clsl +a'bc^l,sl -4a'b\l,sl -a'bl'.sl + 
ab'l^^ + 2a'b\c^llsl - 2a'b\cll',sl + 2ab\llsl + a'bc^llsl - abl^sl - 



a'biy^ +2a'b\c^i:s^ -abl^l^s'^ -abH^s'^ +2a'b\c;^l^s;^ -2ab\l,l^s'^ - 
a'bc^l,llsl + ablUlsl + abllllsl + a'bllsl + ab'llsl - 2a'b\c^l',sl - 
2a'b\cXllsl +4ab\l,iy^ -bc^l'.iy^ -abl'jy^ -abl^iy^ +bc^l,l^iy^ + 



abiy^ -bc^l.iy^ -a'bl^s;^ +2a'b\c^l,'s;^ -2ab\l,^sl^ +a'bc^l,^s'^ - 
ab^l^sl +bc^lll^sl +ablie,sl -bc^lj^l^sl -abl'j^sl +bc^l,l',l^sl -abY.sl + 



2a^b^c^clllsl -2ab'c^l,l^sl -a'bc^lf^sl + abllllsl + abllllsl -abllllsl + 
bc^lilllhl ^abllllsl -bc^ylllsl -4a'b'sl +4a'b\l,sl +5a'b'sl - 
2a'b\c^sl -2a^b\c^sl -4a'b^c^iy^ +ab\iy^ +a'biy^ -ab'l^sl + 



;3_2 



;4„2 „3,,2„2 



2l2. 



/2„2 



;2„2 



ab\l^s;^+2a'bc^l^s;^ -abl^^ -a'biy^ +2a'b\c^iy^ -ab\l,iy^ - 



I'i^i 



l2/2„2 



-3i2„2 



2l2, 



;2„2 



2a'bc^l,iy^ +abl^iy^ -abny^ +2a'b\c^l,\ +ab\lj^s'^ - 



;2_2 



/2;2„2 



2/2„2 



2a'bcJXsi -abl'ltsi +acjn:sl +abl I'si -acJJtltst +a'bi:sl + 



/2i2 2 , „3,,2„2 



''h'^'i'^'h 



i2»5 



'2"5''«>2 



4-5''«k, "^«>|'2-4'-5''«>; 



'■(>''<h 



.2/2, 



;2„2 



abTy^ -4a^b\c^iy^ +4a'bc^l,llsl -ac/.llsl -abl^sl +ac^l,llllsl - 
abllllsl +ac^l,llllsl +abiy^ -ac^l.iy^ -a'bl^sl +2a'b\c^llsl - 
ab'c^ylsl -2a'bc^ljtlsl + abllllsl + abllllsl -ac^l^lsl -abllllsl + 



126 



ac^y!lisl - abHlsl + 2a'b\c^llsl - ablllls^ + abl'j^sl - ablUlsl + . 
abl^llsl -Ua'b'slsl+la'b'c^c^slsl +4a'b\l,slsl +4a'b\l,slsl + 
a'bllslsl +abHlslsl -la^b^c^lls^sl -2ab\llslsl -la^bc^lls^sl + 
abl'^slsl +a'bllslsl -2a'b\c^llslsl+2a'bc^lf,slsl +abllllslsl + 
abHlslsl+2ab\l,llslsl+2a'bc^l,llslsl -abl^l^sy^ -ablll^^^sl - 
a'bllslsl -ab'llslsl +2a'b\c^iy^sl -4ab\ljlslsl -Aa'bc^l.llslsl + 
ablUlslsl+ablUlslsl -abiy^s^ +a'bl^slsl -2a'b\c^l^slsl + 
2ab\ljl^slsl+2a^bc^lf,slsl -abl^l^slsl -abl^l^sy^ +ablie,slsl + 
ab'lislsl+2ab\l,l^slsl +abl^l^slsl -ablXs^sl +abl'Xslsl - 
abl^iy^sl +Sa'b's'^sl -4a'b\l,slsl -4a'b's'^ +4a'b\l,s'^ +Sa'b'slsl - 
4a'b\l,sy^-4a'b'sy^ 

M,,=^a'b'sy^ 

M23 = -4ab'sl si (a' - 4abc^c^ + 2ac^l^ + 1] - ll + 1] - ll ) 

M22 = -b^slsl(a^ + 20a^b^ -Ua^bc^c^ -\6a^bc^l^+4a^c^l^ -4ab^c^l^-2aHl - 
4abc^c^ll + 4ac^ll + /j - 20^/5 +I2abc^c^l^ - Sac^lJ^ - 21^1^ + 1* + 2a^ll - 
Uabc^c^ll + ^ac^lf, + 2llll - 2l^l', + ^ - 2a'l^ + I2abc^c^l^ - 4ac^l,l^ - 
2llll + 2llll - 2llll + /; - 24aVsl - 24aWsl + 24a'bc^l,sl - 4aY,sl 
+ 24a'b'slsl) 

M^i = 2bs^s^{-4a'b'+a'bc^c^ +4a^b'c^c^ +2a'bc^l^-2ab\l^ -SaWc^l^ + 
labc^ll +b\ll -bc^c^l'^+4abHl-2a'bc^c^ll -4abc^l,l^ +a\l,l^ - 
b'^c^hll+c^lHl +bc^c^lt -c^lj', -4abHl+2a^bc^c^ll +4abc^l,l', - 
a\l,l', +b\l,l', -c^llll-2bc^c^llll+2c^l,llll +bc^c^lt-c^l,lt + , 
4abHl-2a'bc^c^ll-2abc^l,ll -b\y^ +2bc^c^l^l^ -c^kllll - 
2bc^c^llll+c^l,llll +bc^c^i:+6aVsl -Sa'b\c^sl +4a'b\l,sl + 



127 



-2i2„2 



-272„2 



2/2„2 



,3r2„2 



.21.3, 



2abXs: -eabntst +6abXs: -6abXst +6a'b'si -Sa'b'c^c^si - 



-2-'^ 



'5"^ 



'6''0, 



S"^ 



'<h 



^ ^ ^ 



4a'bc^l,sl +4ab'c^l^sl +I2a^b^c^l,sl -Aab'^llsl -Aa^bc^c^lls^ - 
eabHlsl +^abc^lj^sl -lallllsl +6ab'l',sl -Sabc^l.iy^ +2allllsl - 
6abH^sl +Aabc^l^llsl -Ga^b'sls^ +Sa^b\c^slsl -UaWc^l^^sl + 



eabHls^sl -eab'llslsl +6abY,slsl) 



M,,=a'b'sl +4a'b'sl -4a'b\c^sl +2a'bY,sl -4ab\c^l',sl +bY,sl - 
2a'bHlsl +4ab\c^l^sl -2bY,l',sl +bY,sl +2a'bY,sl -4ab\c^iy^ + 
2b'lliy^ -2bHlllsl +bY,sl -2a'b'llsl +4ab\c^lisl -2bY,l',sl + 
IbHUlsl -2b'llllsl +bXsl -4a'b's\ +a'b'sl +4a'b'sl -4a'b\c^sl - 
6a'b\l,sl +4a'b\l,sl -4ab\l,sl +2aWl',sl +br,sl +Sab\c^llsl - 
2b\llsl -4ab\llsl +bY,sl -2a'b'l^sl +4ab\c^l^sl +2a'bc^l,l',sl - 
2b\l,l^sl -%ab\l,llsl +4b'llllsl +4abc^c^llllsl -2bc^l',l^sl +bY,sl - 
2hc^hltsl +lllUl +2aVllsl -4ab\c^llsl -2a'bc^lf,sl +2b\l,llsl + 
%ab\l4lsl -4b'llllsl -4abc^c^llllsl +2bc/,llsl -2bHlllsl + 
4K^2^5 ^6 < - 2l',l',llsl + bY.sl - 2bc^l,llsl + I'Xsi - 2aWiy^ + 
4ab\c^lisl -2b\l,llsl -4ab\l4lsl +2bY,lisl +2bY,iy^ - 
2bc^l,llllsl -2bHlllsl +2bc^l,llllsl +bHtsl -a'b^lsl-na'b'slsl + 
4a'b\c^slsl +Sa'b\l,slsl -4a'b\l,slsl +4ab'c^iyy^ - 
ea'b'llslsl -4ab\c^llslsl +4ab\llslsl -bY^s^ +2a'bY,slsl - 
4ab\c^llslsl +Sab\l,l^slsl -2bY,l',slsl -bY^s^^ -2a'bY,slsl + 
4ab\c^llslsl -Sab\l,llslsl +2bY,llslsl+2bH^l',slsl -bYy^s'^ + 
2a'bY,slsl -4ab\c^iy^sl +4ab\ljyy^ -2bY,llslsl -2bY,llslsl + 
2bYXslsl -bXslsl +Sa'b's;sl -4aV4 +Sa'b\l,sl -4aWiy^ + 



Sa'b'slsi -Sa'b\Lslsl +4aWsls' -4a'b'sis 



M3,=^aV.;< 



128 



M33 = 4a^bs^sl(2abc^c^ -labclc^ -b^s^ +2abc^c^sl -Ibc^l^sl -l^^ +1^5^ - 

^n = -a'sl(4aW -^aWc^ -Aab'c^c^ +4ab'clc^ -Aabc^c^ll+Aabclc^ll + 
Aabc^c^ll -Aabclc^l] -Aabc^c^ll +Aabclc^l^ +Aabc^c^l^ -Aabclc^l^ + 
Ua'b'sl +b'sl -Sab\c^sl -Aa'bc^l.sl +Ab\l,sl -I6ab\l,sl - 



2bXs'^ + Abc^lis'^ + ly^ - IbXs'^ + Sabc^c^l^sl - Abc^l^l^sl - 2^1^^ + 
Itsl +2bHlsl -Sabc^c^iy^ +%bc^l,llsl^2llllsl -2lliy^ +/^J -2b'l^sl + 
Sabc^c^l^sl^ -Sbc^lJ^sl -2lll^sl +2lll^sl -21^,1^^ +/,SJ -20a'bX + 
■ 2Aab\l,sl -AbHls^ -AaWsl +AaWc'^sl -l6aWslsl+20aWslsl) 

M3, = -2as^s^{Aa'b'-ab\c^ -"^^'b^lcl -a'b\l, +AaWcll,+2a'bc^l, - 
2ab'c^l^ - a\l', - 2abc^ll + ac^c^t^ - Aa^blj + 2ab^c^c^ll + a^c^l^l] + 
2abc^l,ll-ac^c^ll+Aa'bll-2ab'c^c^ll-a\l,ll +b\l,ll -Aabc^lf, + 
c^llll^-2ac^c^llll -c^lfjl-ac^c^ll+c^yl -Aa'bl^ + 2ab\c^l^ + 
a\y^ -b\l,l^ + Aabc^l,l^ -c^l2l' -2ac^c^lll^ +c^l,lll^ +2ac^c^l^l^ - 
2c^l,lie, -ac^c^l^+c^lj^ -Ga'b'sl +Aa'b\clsl -Sa'b\l,sl - 
Aa'bc^l^sl +Aab\l,sl +Aa^bl^sl +Aab\c^iy^ +6aX'^^ -^^bc^hll^l - 
ea'bllsl + Sabc^lJlsl - 2bl',iy^ + 6a'bl^sl - Sabc^l.l^sl + 2blll^sl - 
ea'b'sl +Aa'b'c,c^sl -AaWclLsi -2a'blhi +6a'blhl -6a'blhl + 



.2l3„2_2 , o„2i.2_ ; 2 1 T„2r;2„2„2 /:„2.t2 2 2 



6a'biy^sl-6a'bl^slsl) 
M,, = a'b'sl -Aa'b'c^clsl +Aa%'cUl -Aa'bcJ^sl +Aa^b'cA,sl +2a'b'cjy. - 



'^^'h"^ 



'<h ^ 



(Ji'i"^ 



'(«l*2''^ 



'«>!'-2''0, 



Sa'b'cll.sl +aY,sl +2aWl',sl +Aa'bc^c^l',sl + Aa' befell', s^ -Aa^bc^l^sl 
2a\llsl ^aY^sl -2aWllsl ^Aa^bc^cHlsl -Aa'bc^l.iy^ -2a\l,iy^ + 
2aY,iy^ +aY,sl +2aWllsl -Aa'bc^cHlsl +^a'bc^l,iy^ +2aV^/,/^J - 



129 






Sa^bc^lJ^sl -la'c^lj^sl +2ab^c^y^sl +4a^l^l^sl +4abc^c^l^l^sl - 



2ac^llljsl + 20^/4/7 5J -2ac^l^lll'^sl -2a^l^lTsl +4ac^l^lll'^sl^ -211iIItsI^ + 
aXsl -2acJ,i;sl +l',l^sl -4a'b'cy^ +Sa'b'cll,sl -Aa'bHy^ +'^a'b'sl + 



a'b'sl -4a'b\c^sl +2a'b'iy^ -4a'bc^c^l',sl +aY,sl -2a'bHlsl + 
4a'bc^c^llsl -2a'llllsl +aY,sl +2aViy^ -4a'bc^c^iy^ +2a'l',iy - 
2aHliy^ + aXsi - 2aWl^sl + 4a'bc^c^l^sl - 2a'l',l^sl + 2aHll^sl - 
2aHll^sl +a'lUl -^a'b'slsl -a'b's^sl +4a'bc^l,slsl -4a'b\l,slsl - 

ea'b'iyy^ +4a'bc^iyy^ -aYyy^+2a'b'iyy^ +4a'bc^i,iy^si - 
2aHiiyy^ -aYyy^ -2a'b'iyy^ -sa'bc^i,iy^si +2aY,iyy^ + 

2aHliyy^ -aYyy^ +2a'bY,slsl +Sa'bc^l,l^slsl -2aY,l^slsl - 
2aYjyy^ +2aHll^slsl -a'l^sX +4aWiy^sl -4a%'sX +4a'b'sX 



APPENDIX B 

CONSTANTS FOR THE FORWARD KINEMATIC ANALYSIS OF THE 

REDUNDANT 8-8 IN-PARALLEL MANIPULATOR 

g^=}-(-a^+ la" a - 2aW -b^ - 2b^J3^ + ll] - odl - ll + cdl +1^) 



q, = i (- a" + 5a^a - SaW + 4a V - 2b^/3^ + Aocb^ p^ + /,' - < + ll - 3odl ) 

2(1 - 2a) 

a, = ^ (fe' - 2o(b^ -4b^l3 + Sccb^P - AaWp + 

' 2b{\-2p + 2aP){\-2a-2l3 + 2apy 

Ab^p^ - 8c*'/?' + 4ar'& V + /,' - < - 2yffl,' + 3ay9,' - a^pl^ - ll + odl + 2/3?' - 

^apll + a^pll + odl - apll + a^pll - apl] + a^pll - afill + a^pll + apll - a^fll + 

apll - a^pll - odl + apll - ^^fis ) 

q, = ? i2b^-4ab^-Sb^p + l6ab^p-Sa^b^P + 

^ 2b{\-2p + 2aP){\-2a-2p + 2apy 

8b^p^ - 1 6ab^p^ + Sa^b^p^ + if - aHf - 2pl^ + 2apll - ll + a^l -h 2pll - 2apll + 

odl + aHl - 2apLl + 2a^pll + all - a^l - 2apll + 2a^pll - ll + 2odl - a^l + 

2pll - eapll + Aa^pil + 1] - 2odl + a^ll - 2pll + 6apll - Aa^pll - all + a^ll + 

2apll-2a^pll-odl-aHl+2apll-2a^pll) 



?6 



-(-2b'^ +9oi}' - 



2b{\ - a){\ -2a + 2ap){\ -2p + 2ap){\ -2a-2p + 2aP) 
UaV + 4a'b^ + 8b^p - AOab^p + 66aVp - AOa'b^p + Sa'b^p - 8b^p^ + 
52ab^P^-96a^b^p^+68a^b^P^-\6a%^P^-\6od)^p'+40a^b^P^- 
32a'b^p' + Sa'b^P' - ll + 3< - 2aHl + 3pll - 1 lapll +\3a^pll - la'pll + 
2a'pil - 2p'll + SaP'll - IQa'P'll + 6a'P'll - 2a'P'll + odl - 3a'll + 2aHl + 



130 



131 



la^pHl - aHl + 2aHl - pll + apll + a^pil - 5a' pll + la'pll + 2J3^l^ - 4aJ3^l^ + 




Qi 



1 



■{p^-Acd)''+Aa'^b 



2 1.2 



2b{\ - a){\ -la + 2ap){\ -2p + 2ap){\ -2a-2p + 2aJ3) 
Ab^p + 1 %ab^p - 24a^b^p + Sa'b^fi + Ab^p^ - 2Aab^p^ + 36a^b^p^ - \6a'b^p^ + 
Sab^P' -Xea^b^P' + Uc'b^p' + al^ - 3a^l^ + 2a'l^ - 2apif + 6a^pl^ - Aa'pl^ + 
2apH^ - ea^pH^ + Aa'pH^ - all + 3aHl - 2a'll - Aa^pl^ + Aa'pll + 2apHl - 
2a^pHl + ll - Soil + 9aHl - 6a' l^ - 2 pi] + \Aapil - 30a'/?/' + 2^a'pll - 
%a'pll - 6apHl +\Za^pHl - 20a' pH^ + Sa^p^l^ - 1] + Aal^ - SaH] + 2a'll + 
2pll - \Oapll +\6a^pll - Sa'pl^ + 2apHl - 6a^pY, + Aa'pH^ + all - ^oc^^l + 
2a'll - 2apll + 6a^pll - Aa' pi] + 2apHl - 6a^P^ll + Aa'pH] + all - ^cc^^l + 
2a'll - Aapll +\6a^pll - 2Qa'pil + ^a'pil + 2apHl - lOa^P^ll + \6a' p^l - 
Sa'P^ll - all + Sa^ll - 2a'll + 2apll - 6a^pll + Aa'pll - 2apHl + 6a^p^ll - 
Aa'pHl - aHl + 2a'll + 2apll - Aa^pil - 2apHl + 6a^pHl - Aa'pHl ) 



?8 



■(^'-4o*'+4a'fo 



2r2 



2b{\ - a){\ -2a + 2ap){\ -2p + 2ap){\ -2a-2p + 2aP) 
Ab'p + \%od)^p - 2Aa^b^P + %a'b^ p + Ab^p' - 2Aab^p' + 36aW p^ - I6a'b^p^ + 
Sab'p' -\6aWp' + Sa'b^p' + a'lf - 3a'll + 2aUl - 2apll + Aa^pil - 2a' pll + 
2apHl - 6a'P'll + Aa'p'll - a'll + 3a'll - 2aUl - 2a'pil + 2a'pll + 2apHl - 



132 




4aHl-2a) 



Aa\\-2a){b-0dj-q,-lq,+20(q,) 



9 2 

3a^/,^ + /j - 40/3' + 5a^l^ - lo'l] + odl - 3a^l^ + 2ar'/6 + 1^ - lodj + a'^lj + 
2bq^ - %0d?q^ + IQa'^bq^ - Aa^bq^ - 4bj3q^ + llocbj^q^ - Sa^b/Jq^ - la^q] + 
Aa'ql - 6abfiq^ + I4a^bfiq^ - Sa^bfiq^ + 2aq^q^ - Aa^q^q^ - 4bq^ + l2oi)q^ - 
lOa^bq^ + Aa^bq^ + Safcyfi^^ ~ 24«^&/5^6 + lea^bfiq^, + Aq^q^^ -l2oij^qf, + 
Sa^qsqf, + 2ql - Saqj + lOa^qj - Ao^q] + 20*^7 - 6a^bq^ + Aa^bq^ + 2b pq^ - 
\2(Xbpq^ + 22a^bpq^ - I2a^bfy^ + 2q^q^ - 6oaj^q^ + Aa^q^q^ + 2bq^ - Aabq^ + 
2bpq^ - Sabfiq^ + lAa^bfiq^ - Sa'bfiq^ - 2q,q^ + 6aq,q^ - Aa^q^q^ ) 

9 -^ 

A^, , = -^— (a^b ~Aa^ab + Sa^a^b - 2a^a'b + bl^ - 2ablf + a'^blf - 2otbj3l^ + 

2a^bj3lf + 2abpi^ - 6a'bfi^ + Aa'bfil^ - 2abl3ll + 6a^bpll - Aa'bpll - bll + 
2(xbll - a^bll + 2ocbl3ll - 2a^bpll + b\, - 3o(b\, + 2aWq^ + 2b^ pq^ - 
eocb^pq^ + AaWpq^ + I'^q^ - 2odfq^ + a^lfq^ - 2l^q^ + lod^q^ - la^lq^ + 
2a'llq^ + llq, - Aodlq, + Sa^lq, - 2aHlq, - od^q, + aHjq^ - 2bql + 6odjql - 



133 



4a^bql + 4abPql - Sa^bfiq] - la^oeq^ + Sa^a^q^ - lOa^a^q^ + 4aWq^ + 
lodj^Pq, - 2aWj3q, + Ib^p^q, - lOob^fi^q, + SaWfi\ - 2lfq, + Aod^q, - 
la^l^q^ + lodlq^ - 2aHlq^ + Ibq^q^ - 6ccbq^q^ + 4a^bq^q^ - Ibpq^q^ + 
4cxb/3q^qs + 2a^b/3q^q^ - 4(Xbfiq^^ + 4a^bj3ql - 4a^q^ + ISa^O^g - ?>Qa^a^q^ + 
24a'a'q^ - SaWq^ - 2b'' q^ + 5(Xb'q^ - 2aWq^ + 4ody'fiq, - Sa'b'/3q^ - 
4b'/3\ + i2c(b'P\ -^a'b'j3\ -od^q,+a'l^q,+2l^q,-6od^q,+5a'l^q, - 
2a'l^q, + Cdlq, - 3aHlq^ + 2a^llq, + 2llq^ - 6odlq^ + 5aHlq^ - 2bq,q^ + 
6abq,q, - 4a'bq,q, - 4ocbfiq^q, + Sa'bj3q^q, + 2ab/3q,q^ - 6a'bj3q,q, + 
2aq,q,q, - 4a'q,q,q, - 2bql + 40!bql + 4cxbfiql - Sa'bpql + 2q,ql - 6aq,ql + 
4a'q^ql +a'q-j -Sa^oq-j +I0a'a'qj -lOa'a^q^ +4a'a*q^ -4od?'fiq^ + 
na'b'pq^ -Za^b'pq,+2b'p\, -2od)'p'q, -Sa^b^fi^q, +%a^b'p\, - 
l^q^ + 'iodlq-^ - 2a^/f ^7 - llq^ + liCdlq^ - 2a'l^q-, + 2bq^q^ - Sabq^q-j + 
lOa'bq^q-j - 4a^bq^q.j - 6b/3q^q^ + 22od}fy^q^ - 22a'bPq^q^ + 4a^bl3q^q^ - 
2aqlq, + 4a'qlq^ - 2(Xb/Jq,q^ + 4a'bpq,q, + 294^5^7 - 4aq,q^q, + 2abq,q, - 
6a'bq^q-j +4a^bq^q.j +4bfy^q^ -i^odyPq^^q.^ +lSa'bfy^q^ -4a^b/3q(,q^ + 
2^496<?7 -6«?W697 +4a'q^q^qT+a'q^-5a^aq^+\0a^a'q^-l0a'a^q^ + 
4a'a'q^ + b\^ - 2ab'q, - 2b^j3q^ + 6a'b'pq^ + 2b'fi'q^ - 2ab'fi'q^ + 2odfq, - 
2a'lfq, - 2l^q, + 5od^q, - 2a'l^q, + 1^ - ^ocl^^s + 2a'llq, - l^q, + 2od^q, - 
2aHlq^ + 2bq^q^ - 6abq^q^ + 4a'bq^q^ - 2bj3q^q^ + 6abJ3q^q^ - 6a'b/3q^q^ - 
2aqlq, + 4a'qlq, + 2bfiq,q, + 4bpq,q^ -\6o(b/3q,q, +\M'bpq^q^ - 2q,q,q, + 
6«S?4^698 - ^0C^<l4<l6^» + 2bq^q^ - 6oi>q^q^ + 4a^bq^q^ - 6bpq^q^ + 2QiOi>pq^q^ - 
I6a'bj3q^qs + 2bql - 4cxbql - 2bj3q^ + 4(Xbj3q^ - Aa'bpq] ) 

yV,o = a'(- a' + 4a V - 5a'ar' + 2a' a" - a'b' + 3a'Q*' - 2a'a^b' - 3a'lf + Sa'od^ - 
laWlf + 2aWl^ -b'l^ +cxb'l^ - 2b^J3'l^ + 2ab'/3'lf + 2aHl -9a'od^ + 
XSa^a^ll -Ua'a'll +4a'a'll +2b' PHI -eceb' PHI +4aWj3'l', +cdfl^ - 
aH'^ll - aHl + 6a'odl -UaWll + \2a'a'll - 4a'a'll - 2b'pY, + 6ab'/3'l', - 
4a'b'pHl + l^ll - 2(xllll + aH^ll + 2a'll - 5a'od', + 5aWl^ - 2aWll + bY, - 



134 



ccbHl + Ib^pHl - lab'fiX + IX - cdfls - lllll + 50^3 ^8 - 3a'/3 ^8 + ^l^l - 
Aodlll + 3aHlll - /; + Odt - Aa^bpq, + 1 la^dbpq, - SaWb/5q, - 2b^ pq, + 
Aoi)'l3q, - IbplU, + 2od)pl^q, + labpllq, - Aa'bpllq, + Ibpllq, - 6c(bj3l^,q, + 
Aa^bpllq, + locbpllq^ + Aa^ql -\6a^aql + lOaWql - SaWql + Ib^q] - 
4ceb'ql - lllql + Aodlq] - lllq] + Aodlql - 2a'b/3q, + 4a^ab/3q, - 6aWb/ii, + 
4aWbPq, - lodj'pq, - 4b'/3'q, + 4ab'fi'q, + 2bj3lfq, - 2cd?j3l^q, + 2a^bfil^q, + 
2(Xbpllq^ - 2a^bl3llq^ + 2bpllq^ - Aocbfil^q^ - 2a^q^q^ + lOa^oq^q^ - \6a^a^q^q^ + 
SaWq^q,-2b^q^q, +4(Xb^q,q, -4od)^fi^q,q,+4l^q,q,-l0odlq^q,+4aH^q,q, - 
2llq,q,+6odlq,q, -4aHlq,q, + 2llq,q, -4cdlq,q, +4bfiqlq, -Sabfiqlq, + 
4ab^P^ql -4bpq^ql+%0i>Pq^ql + 2a^bq^ -\0a^Ocbq(,+\6aWbq^ -Sa^a^bq^ + 
4b'fi\ - Sab'fi'q, - 2blU, + 2od>l^q, - 2bl',q, + 6otbl',q, - 4b'j3q,q, + 
Sod?'^,q, - 4odj^fiq,q, + 4bq,q,q, - 8(Xbq,q,q, + 4a^ql -IGaWe + 20aWql - 
SaWql + 2b' ql - 4(Xb'ql + 4b'fi'ql - Sc(b'/3'ql - 2l^ql + 4od^ql - 21^ ql + 
4cd^ql -2a'bq.j +\2a'od?q^ -26a^a'bq.j +24a'a^bq^ -Sa'a'^bq^ +4a'b/ij^ - 
\8a'ocb/3qT +30aWbJ3qT -24aWbfiq^ +Sa'a'bj3q^ -4b'fi'q,+\2(xb^J3'q, - 
Sa'b'p'q^ +4b^fi\,-\2ab^p'q, +8a'b^j3^q, +2bl^q, -4o(bl^q, +2a'blfq^ + 
2abfl^q^ - 2a'bpl^q^ + 2bllq, - %oi>llq, + 6a'bl^q^ - 4bj3l^q^ + lOoibpl^q^ - 
ea'bpilq, ■\-4b'pq^q,-\2(Xb'pq,q, +8aWfiq,q, +4ocb'[5\,q, -8a'b'/3'q,q^ - 
4bpqlq, +^CCbpqlq,+4cd)''l3q,q, -4a'b'Pq,q, +4a^b'fi'q,q, -4bq,q,q, + 
l2oi>q^q^q^ -Sa'bq^q^q^ +Sb/3q^q^q^ -20oi>fiq^q^q-j +8a'bfiq^q^q.j -2a'qf^q^ + 



Wa'aq^q^-ieaWq^q^ +Sa'a'q^q, -4b' P'q^q^ +'&ccb' P\^q, +2l^q^q^ - 
2cdU(,Qi + 2/8^697 -^cdUt^li+^bpq^q^q^ -8o±>Pq^q^q^ +4od?fiq^q^q^ - 
4^4^5^697 +^Otq,q,q^q, -4bpqlq, +8abfiqlq, -2a'bq,+\0a'abq^-l6aW^^^ . 
Sa'a^bq^ + 4a'bpq^ -XQa'ocbpq^ +\0a'a'b/3q^ -4a'a^bPq^ + 2b^ Pq^ - 2ab^ pq^ - 
4b^p'q, + Sab'P'q^ + 4b'p'q, - 4od?^p'q^ + 2bl^q, - 2c(bl^q, + 2bpi^q, - 
2cxbpl'q^ - 4bpl^q^ + lOabpl'q^ - 6a'bpllq^ + 2bpl^q^ - Socbfil^q^ + Ga^b/Jl^q^ + 
2bllq, - eccbllq, - 4bpilq, + 4(xbpilq, + 4b' fiq.q, - 8o(b'j3q,q, + 4od?'j3'q,q, - 
4bpqlq^ + Sotbfiqlq, + 4ab'j3q,q, + 4b'fi'q,q, - 8ab'fi'q,q, - 4bq,q,q, + 



135 



Sabq^q^q^ + ^bpq^q^q^ - Sobpq^q.q^ - la^q^q^ + Wa^Oq^qg -\6a^a^q^q^ + 
SaWq.q, - 2b^q,q, + 4od?'q,q, - 4b^fiq,q, + \2ab^/3q,q, - 4b^/3\q, + 
Sab^fi'q.q, + 4l^q,q, - \0(Xl^q,q, + ^aH^q^q^ - 2llq,q^ + 6cdlq,q^ - Aa^lq^q^ 
2^8 <7698 - 4crf8 96^8 + ^bpq.q^q^ - Mbpq.q^q^ - 4bfiq,q^q^ + %od}pq^q^q^ + 
Abqlq, - Socbqlq, - 4bfylq, + ^obfiqlq, - 4a^aq^q, +\2aWq,q, - SaWq.q, - 
4b'j3q,q,-\6(Xb'j3q,q, + l2a'b'j3q,q,-4b'fi'q,q,+l2ab'fi'q,q, - 
Ua^b^fi^q^q^ -4lfq^q^+4odlq^q^+4cdlq,q^ -Sbpq.q^q^ +\6ab/3q,q^q^ + 
4qlqiqi -^0(qlqiqi-'^Oi>lk5Qi(i%-^bq^(ii^^'^\2oi)q^q,q^ -%a^bq^q^q^ + 
l2bfiq,q,q,-32c(bfiq,q,q,+Sa'b/3q,q,q,+4b'fiql-\2odj'fiq', -4b'fi'q', + 
40(b'P\l - 4bq,ql + Mbq.ql + 4bPq,ql - Sabfiq.ql + S(Xbj3q,q', ) 



+ 



N..= 



8fl^a(l - 2a) 



^23 = -^^-^—^{b -oi)- 4abJ3 + 4a^bp + 20" q^ +aq,+ 2aq^ - 2a^q^ +q,-aqT + 

(I ~ OC) 

N,, = ^^(-2a^ +16a^a-46aV +60flV -36aV +8aV +&' -5o*^ + 

8a'&' - 4a'b^ - 4b^fi^ + 16c*'/?' - 2QaVp^ + 8a'&'y9' + /,' - 3< + 4a'/,' - 
2a'/' + 2/3 - 8crf' + lOa'/' - 4a'/' - ll + 5cdl - %aHl + 4a'/,' + 2/g' - \Qodl + 
Ha'/g' - 6a'/g' - 66^4 + 24c*^4 - 30a'^94 + Ua^bq^ + 4b pq^ - %(xbpq^ - 4a^bpq^ + 
Sa^bfiq^ - 2bq^ + Sodjq^ - lOa^bq^ + 4a^bq^ - 2b/3q^ + \4odj/3q^ - 24a^bfiq^ + 
\2a^bfiq^ - 2a^q^q^ + 4a'94^5 + 2/7^6 ~ lOofe^g + Ua'/?^, - 4a^bq(, - locq^q^ + 
6a'^5^6 - 4a^q^qf^ + 6oi)q^ - 1 Sa^bq^ + 1 2a^bq^ + 4bPq^ - 20cdj/]q.j + 32a^bfiq^ - 
\6a^bPq^ -6aq^q-i +\Sa^q^q., -Ua^q^q^ -2q^q-, +6c(q^q-, -4a^q^q^ -2qf^q.^ + 
Saq^q^ - lOa^q^q^ + 4a^qf^q^ - 6bq^ + 22oi>qg - 22a^bq^ + 4ar'bq^ + 4b Pq^ - 
\6cd)pq^ + ^6a^b/3q^ - 4a^b/3q^ - 2aq^q^ + 6a^q^q^ - 4a^q^q^ - 2q^q^ + Saq^q^ - 
\0a^q^q^+4a^q^q,) 



T^r^' 



136 



N., = 



la' 



'' \-a 



[a'b -4a^ab + 5aWb - la^a'b + 2b' - 6ab' + 4a^b' - Aa'abp + 



l6aWbj3 - lOaWbfi + SaWbfi + 2bl^ - 4abl^ + la^blf - 2oi>pl^ + la^bpl^ - 
ml + \Oabll - 9a^bll + la'bll + lod^pll - ba^bpll + Aa^bpl] + bll - AMI + 
5a^bll - la'bll - Icdjpll + 6a'bJ3ll - Aa'bpll - IMl + la^bll + loebpll - 
la^bl3ll-2a^q^ + \0a^Ocq^-\%a^a^q^ + \Aa^a\^-Aa^a\^-3b^q^ +9ab^q^ - 
ea'b'q, + 2b'fiq, - 60d?'/3q, + Aa'b'fiq, - Ab'fi'q, + Uocb'fi'q, - Ba'b'fi'q, + 
3llq, - 9odlq, + 6aHlq, + llq, - Sod^q, + 2aHlq, - a\, + 6a^0Cq, - l3aWq, + 
Ua'a'q, - Aa'a'q, - b\ + 3(Xb'q, - 2a'b'q, + 2b'fiq, - 2odj'fiq, - 2b'fi'q, + 
2ab^P\^+2llq5-l0dlq^+la^llq^ -2aHlq^ -llq,+Aodlq, -Sa^l^q^ + 
2aHlq, + llq, - 3odlq, + 2aHlq, - Abq.q, + \2(Xbq,q, - Sa^bq.q, + 2bfiq,q, - 
2c(bpq^q, - Aa^bpq^q, - 2b pq] + 6odjpql - Aa^bpq] + 2a^(Xq^ - Sa^a^q^ + 
lOfl V^e - 4a^«'96 - 2^^^6 + 5a^ ^6 " 2a^b^q^ + Aab^ pq^ - %aW pq^ + od^q^ - 
2aHlq^ -Odlq, + 2aHlq, + 2bq,q, - 6abq,q, + Aa^bq.q, + 2a'q, -9a'aq, + 
ISa^a^q^ - \2aWq^ + Aa^a^q, + 3b^q^ - 9oi)^q^ + ^Wq^ - Aa'b^q^ - 
6b^pq^ + leab^pq^ - 6a'b^pq^ - Aa'b^flq^ + 2b^P^q, - 2ab^P^q, - 
Sa^b^P^q^ +Sa'b^P^qT+2odfqT-2a^l^q^ -l^q^ +5aHlqT -2a'llq^ +od.lqn - 
Sa^l^q^ + 2aHlq, -llq, + 3odlq^ -Aa^lq^ + ^bPq.q, - \6abpq,q, + 
Sa^bpq^q-j - Ibq^q^ + Sobq^q-j - lOa^bq^q^ + Aa'bq^q^ + Abpq^q^ - \2od)pq^q^ + 
lAa^bPq^q^ -Aa'bpq.q, -Aaq.q^q^ +Sa^q^q^qT +2bq^q^ Sotbq^q^ + 
Sa^bq.q^ - 20CbPq^q^ + Aa^bpq^q-, - 2q,q^q, + 6aq,q,q^ - Aa^q^q^q^ + 2abq^ - 
6a^bq^ + Aa'bqj - 2bpq^ + Aoi>pql + 2a^bpq^ - Aa'bpq^ - 2q^q] + 6aq^qj - 
Aa^q^q^ +a^C(q^-5a^a^q^+%a^a'q^ -Aa^a^q^ -cd?^q^ + 2a^b^q^ -Sod^^pq^ + 
UaWPqs - 2b^p^q^ + lOob^P^q^ - Sa^b^p^q^ + 2llq^ - 6odlq^ + 5a'^llq^ - 
2a'llq^ + Cdlq^ - 3aHlq^ + 2aHlq^ - OCl^q, + 2aHlq, + 2bpq,q, - ^Obpq^q^ + 
Sa^bPq.q^ - 2bq^q^ + 6od)q^q^ - Aa^bq^q^ + 2bpq^q^ - Aobpq^q^ - Abq^q^ + 
lOabq^q^ - Aa^bq^q^ + 20!bPq^q^ - Aa^bpq^q^ + 2bq^q^ - 2abq^q^ - 6a^bq^q^ + 
Aa'bq-^qg + 2bpq^q^ - lOoirpq^q^ + lOa^bPq^q^ - Aa'bPq^q^ - 2q^q^q^ + 
^Oiq.qjq, - Aa^q.q^q, + 2abql - Aa^bql - 2ocbPql + Aa^bpql ) 



137 



iVjo = a'(-2a' + 14aV-38flV +50flV -32aV +8flV -aV +3a'a*' - 
2aWb' -b' + 2c(b' -Aa'b'fi' +I6a'ci>'fi' -20aWb'fi' +SaWb'fi' - 
2b'lf + 2(xb'l^ - 2b'fi'lf + 2cd>^fi^lf + 3aHl -Ua^odl + 2Qa^aHl - lAa^aH] + 
Aa'a'll + 3b^ll - lotb'l^ + 2aWl^ + 2b'fi'l^ - 6(xb'fi'l', + Aa'b'p'll + 2lfl^ - 
2cd^ll - 2ll + 5< - 2aHt - 2aHl + IQa'all -l^a'aHl + lAa'a'll - Aa'aY, - 
b'll + 3(xbY, - 2a'b'll - 2b'l5Hl + 6(xb'fi'll - 4a'b'/3'l', + l^l - 3od^l', + 
2aHlll + ZaHl -\3a'od^ + ISaWl', - SaWl^ + 2od?X + ^b'fi'l^ - 2ab'P'l', - 
lUl + odlll - 2aHlll + llll - 3odlll + 2aHlll - it + 2< + Aa'bPq, - 
\6a^(Xbpq, + 20aWb/3q, - SaWbj3q, - 2b'j3q, + Aab' pq, + 2bpllq, - 
Aab/3l^q,-2bfil^q,+Aabj3l^q,-Aa^bj3q, + l6a^odjfiq,-20a'a^b^,+ 
%a^a'bPq, - 2b' Pq, - Ab'P'q, + Aab'P'q, + 2bpllq, + 2bpilq, - Acxbfil^q, + 
Ab'q.q, - %oi>\,q^ + Ab^ fi^^q, - Sab^p\,q, - Al^q.q, + Sod^q.q, + Aa^bq, - 
16a^abq^ + 20aWbqf, - Sa^a'bq^ + 2b' q, - Aob'q^ + Ab'fi^q^ - Sab'fi^q^ - 
2bllqf^ + Accbllq^ - 2bl^q^ + Accbl^q^ - Aa^bq^ + 2Qa^od)q^ - 36aWbqj + 
2SaWbqy -Sa^a*bq^ -2b'qT + 6ccb'q^ -AaVq^ ^ea^bpq^ -2(ia^(Xbl3q^ + 
A0aWb/3q^ - 2Sa^a'bfiq, + Sa^a'b/3q^ + 6b'fiq^ - lAodj'fiqj + Aa^b'fiq, - 
Ab'P'q,+l2c(b'fi^q,-^a^b'fi'q,+Ab'fi'qT-\20Cb'l3'q,+Sa'b'fi'qT + 
Abplfq^ - Aabpllq, + 2bllq^ - 6ccbllq, + Aa^bl^q^ - Wpilq, + 20abpi^q^ - 
%a^bpllq^ + 2bpllq^ - 6(Xbpllq^ + Aa^bfil^q^ + 2bl^q^ - 6ocblgq^ + Aa^bl^q^ - 
2bj3l^q^ + 2od)pllq, - Aa^bpllq, + Ab^j3^q,q^ - %ab^p^q^q^ + Ab^/3^q,q^ - 
Wpq,q,q^ + I6abfiq,q,q, - Aa'q^q^ + I6a^aq^q^ - 20a^a^q^q^ + Sa^a'q^q^ - 
2b\^q, + Adb\s, - Ab^Pq^q^ + Sob^fiq.q^ - Ab^fi'q.q, + &od?'fi\q, + 

2/3^6^7 - 4crf'9697 + ^^ke^ii - ^odkeQi + 2fl'^7 " lOa'os?' + 16a'a'^7 - 
Sa^a'q^+Ab^/3q^-l2ab^/3q^+Sa^b^/3q^j -Ab^jS^q^^ +l2od?^j3^q^ - 
Sa^b^fi^q^ - 2lfq^ + 2cdfq^ - 2l^q^ + 6od^q^ - AbJ3q,q^ + 2>(XbPq^q'^ - 
AC(bj3q,q^ +Aq,q,q'^ -Soq.q.q^ + Abj3q,q^ -^(xb/3q,q^ -Aa^bq,+l6a^od?q, - 
20a^a^bqg + Sa^a^bq^ - 2b' q^ + Aoi}'q^ + 6a^bpq^ - 26a^od}pq^ + 
36aWbfiq^ - lea'a'bfiq^ + Aab'fiq^ - Ab'fi^q, + Socb'fi^q, + Ab'fi'q^ - 



138 

2 _ 



eod^pilq, + Aa^bpilq, + Ibllq, - 4abl^q, - Abpijq, + Sod?j3l^q, - 4b' P\, 



SO(b'/3'q,q,+4b'P'q,q, -Sccb'fi'q,q,+4b\q, Sob'q.q, -4b'/3q,q, + 
^ab'j3q,q,+4b'fi\q, -Sab'fi\q, -4l^q,q,+Sod^q,q,-2a'q^q, + 
XQa'aq^q^ - Iba'a'q^q^ + SaWq^q^ - Ib'^q^q^ + 4ab\^q^ + Sb'fiq^q^ - 
lOab'pq^q, +Sa'b^fiq,q, -Sb'j3'q,q,+l2c(b'P'q,q, -Sa'b'j3\^q,+4l^q,q, - 
lOod^q.q, + 4aHlq,q^ - 2l^q,q, + 6cxl^q^q, - 4a'llq^q, + lljq.q^ - 4od^q,q, + 
4bpq,q,q^ - Sab/3q^qjq, - 4bPq^q,q^ + %cA)fiq^q,q^ + 4bq^q,q^ - 2>abq^q^q^ - 
^bpq(,q-,q^ + Sab/3q,q^q, - 4bq^q, + llocbq^q^ - ^a'bq^q, + 4bPq^q, - Sobfiq'q, + 
Sa'bflq'.q, + 4b'/3ql - Sc(b'fiq', - 4b'[3\l + ^0(b' P\l - 4bq,ql + m?q,ql + 

.,0 _2 o„.i.fl_ _2\ 



_ 8a'ar(l-2a) 



iV^^ = _ 4a (1 2a) ^^ -3c(b + 2a'b - 4abp + 4a'bJ3 - 2a\, + 3aq, + 20^^ - 2a\, + 
q^-aq^+q^-aq^) 

^ 2a\\-2oO ui _ j2^2^ ^ 22a'«' - 16a'a^ + 4fl V + &' - 3£*' + la'b" + 
'' (1-a)' 

4fe'y9' - 8c*'y9' + 4a'&'y9' - 1] + ai,' - 2/3' + 60^3' - (id'l] + 2a'/3' + ll - Zcdl + 

4a''ll - la^l - 2ll + 4cdl - la^l - 2bq^ + 4abq^ - la'bq^ - 4bj3q^ + I6(xb/3q^ - 

1 2a'b/3q^ + 4bq^ - Sabq^ + 4a' bq^ + 2b Pq^ - 1 %dbpq^ + 1 6a'b/5q^ -2a'q^q^ + 

2aql - 26^6 + 2c*9g - 2a'bq^ + 6aq^q^ - Ga'q^q^ + 2bq^ - 6od)q-, + Sa'bq^ - 

4a^bq.j -4bPq^ + 8abfiq.j - Sa'bfiq^ + 4a^bpq^ - 2aq^q^ + 2a'q^q-, + 4q^q^ - 

4aq,q, + 2q^q, - 4aq,q, + 2a'q,qy + 2abq, - 4bfiq, + Uabfiq, - Sa'b/3q^ - 

6aq,q^ + 6a'q^q^ + 2q^q^ - 2aq^q^ + 2q^q^ - 4aq^q^ + 2a'q^q^ ) 



139 



w , = ^^(a'^ - 6a'fl* + na^a^b - Ua'a'b + 4aWb - Aa'obfi + Xea'a^bp - 

20a^a'bp + SaWbfi - blf + 2ablf - a^bl^ + labfil^ - la^bpL^ - loiyfil + 
ea^bpll - Aa'bpll + Kxbpil - Ga'bffll + Aa'bfil^ + blj - loi)ll + a^bll - 
locbpll + la'^bpll + 20^4 - 10a'a94 + \SaWq, - Ha^a'q, + Aa'a'q^ + b^q^ 
3(xb\ + 2a'b\-2b'fiq,+6c(b'Pq, -4a'b'j3q,+4b'fi\-l2ab'fi\ + 
SaW/3\, - l^q, + 3odlq, - 2aHlq, - 3llq, + 9(xllq, - 6aH^q, - 3a'q, + 
ISa'o^s -39aWq^+36aWq, -na^a'q,-2b\^ + 6ab^q^ -Aa^b'q^ - 

5 + 2aWPq, - 6b^fi^q, + 22ab^j3\ -\6a^b^fi\ + ifq, - 2alfq, 

\-4l'; 
+ 3, 

22abfiq,q,- , ,_, ., ., 

\Sa^bfiql + 2a^aq^ -SaWq^+lOaWq^ -4a^a*q^ +ab^q^-2aWq^ - 
4ab'/3q, + Sa'b^j3q, - al^q, + 2aH^q, + Od^q, - 2aHlq, + 4bq,q, - I2(xbq,q, + 
%a'^bq^qf,-2aqlq(^+4a^qlqf, +a^aq.j -Sa^a^q^ +Sa^a^q-j - 4a^ a* q^ + 4od}^ pq^ - 
\2aWl3q, +%a'b^pq, -2b^fi^q,+2ody^fi'q, +SaWp^q, -Sa'b^J3\^ -od^q, + 
aHlqT+2llq^ -Sodgq^ +3a^lsqT -2bfiq^q^ +4abpq^q^ -4bq^q^ +l6odjq^q^ - 
20a^bq^q^ + Sa^bq^q^ + Sbfiq^q^ - 2Sody/3q^q^ + 26a^bpq^q^ - Sa^bfiq^q^ + ^ 
2aq,q,q, -4a\,q^qT -2q]q, +4aqlq, +20(bl3q^q, -4a^bPq^q, -4q,q^q, + 
12095^6^7 -Sa^q^q(,qj +2a^q^ -9a^(Xq^+\5a^a^q^-\2a^a^q^+4a^a*q^ + 
b'^q^ - 4oi)\, + 4a^b\^ + 2b^j3q, - 6a^b'j3q, + 2b^p^q, - lOab^fi^q, + 
SaWj3'q, - Odfq, + a^l^q, - l^q, + 6od^q, - lOa^.q, + 4a'l^,q, - 2odlq^ + 
eaHlq^ - 4aHlq^ - llq^ + 3odlq^ - aH^q, - 6bj3q^q, + 2Qoibpq^q^ - lea^bpq.q^ - 
4bq,q^ +I2abq,q^ - Sa^bq.q, + 6b/3q^q, - 24c(bfiq,q^ + 20a^bfiq,q^ + 2aq^q,q^ - 
4a^q^q^q^ + 2^96^8 " 2«*^6^8 " ^OC^bq^Qi " 2a*^6^8 + ^^C^bpq^q^ - 2q^q^q^ + 
^O&is^i^s - 4^^959698 - 4«f'97^8 + i2a^bq^q^ - Sa^bq^q^ + 2bpq,q^ - 2od)pq,q^ - 
6a^bpq,q^ + Sa'bfiq^q^ + 2q,q,q^ - 6c(q,q,q^ + 4a\^q^q^ - 2bql + 2oi}ql + 
4a'bql - 2bPql + lOcd^Pql - lOa'bPq', + 2q,ql - 6aq,ql + 4a'q,ql ) 



140 




^^, ^.^,+4aWbPq, + 2cd?'fiq,+4b'j3'q, . ,^ 

labpllq, - ^bpllq, + eabpllq^ - Aa\,q^ + \6a^aq,q^ - 20aWq^q, + 
Sa^a'q,q,-2b\,q,+4ab\q, -4b^/3\,q, +%(Xb^ p\,q,+lllq,q, - 
Acdlq^q, + 2llq,qs - ^cdlq^q, + 2a^ql - XQa^CXq] + \6a^a\l - %a^a'q] + 
2b' ql - Aoib'ql + Ab'p'q] - Sod^'fiY, - AlWs + XOoilq] - Aa^Ws + 2llql - 
6odlql + Aa'llq] - 21^, + Aod'.ql - AbPq.q] + Sobfiq.q', + Ab/Jql - ^ab^q] - 
Ab'P'q, + Socb'P\ + Abllq, - Sabl^q, - Abqlq, + ^ocbqlq, + 2a''bl3q, - 
l0a'od?fiqT+l6aWbPq^ -SaWbfiq, +Ab'p\^ -I2ceb'fi'q, +Sa'b'fi'q^ - 
Ab'fi'qj + \20Cb^P\, -%a'b^p\^ - 2bpllq^ + 2abfilfq, -Abl^q, + \2od)llq, - 
Sa'bl^q^ +6bl3l^qT -lAabfil^q^ +Sa'bj3l^q^ -Ab'' P^^q-^+^OCb' p'q^q^ - 
A0(b'/3\,qj+Abfiq,q,q^ -Sobpq.q.q^ +Abq',qT -I2(xbqjq, +Sa'bq',q^ - 
%bPq]q, + 20ctb/3qjq, - %a'bl3q]q, + Ab'fi'q.q, - Sccb'fi'q.q, - Al^q.q, + 
SCKfg 96^7 + Mke^li - ^OCqU(,Qi + ^a^b^q^ " ^Oa'obfiq, + \2aWb/3q, - 
AaWbJ3q, - 2b' Pq, + 2ab'pq, + Ab'fi\ - Sab'fi'q, - Ab'fi'q, + Aob'fi'q, - 
2bj3lfq^ + 2oijpl^q^ + 6bpllq^ - lAob^^q^ + Sa^b/Sl^q^ - Abfil^q^ + l2ocbj3l^q^ - 
%a'bpilq, -Abllq,+^CCbllq,+Abpilq, -Aabfillq^+Ab' P\,q, -Sab'fi^q, - 
Sb'fi\q, + Uab'P'q.q, + Abj3q,q,q, - ScxbPq,q,q, + Abqlq^ - Sobqjq, - 
Abpqlq,+S(Xbpq',q, -Aa'q,q,+l6a'cxq,q,-20a'a'q,q,+%a'a'q,q, - 
2b'q,q,+Acxb'q,q,+Sb'fiq,q,-l6ab'fiq,q, -Ab'/3'q,q,+Sab'j3'q,q, + 



141 



SaWq^q, -Sb'Pq,q,+24ocb'fiq,q,-l6a'b'j3q,q,+Sb'/3'q,q,-20ab'j3'q^q, ■ 




4a\l-2a) , ,_ ... , . 

N,,=-^^-^^^^{b-(xb-2abfi + 2a'bfi + aq,+q,-aq,) 
(l-ay , . 

A^,, = ^^(2a'«-10aV +18aV -14aV +4aV -c*' +3aV -2a'&' - 

'' (1-a)'^ 

2b^fi^ + Sotb^fi' - WaVfi' + 4a'b^/3' + if - 3cdf + AaHf - laHf + 1] - 4od^ + 

Sa^l - la'll + oil " ^X + 2a'll - lodj + AaX - laHl - 2^94 + ^oijq, - 

lOa^bq^ + 4a' bq^ + Ibpq^ - 6abPq, + 4a^bpq^ - a^q] + 2a' q] + 4abJ3q, - 

Sa^bPq, + 4a'bPq^ - 4bq, +\2abq, - lOa^bq^ + 4a' bq^, + 4abPq, -\2a^b/3q, + 

Sa'bfiq, - laq.q, + 6a\,q, - 4a'q,q, - ql + 4aql - 5a\l + la'q] + 2bq, - 

4ocbq, - 2a^bq, + 4a'bq, + 2b pq, -\2oi}Pq, + 22a^b/3q, - \2a'b/3q, - 2q^q^ + 

6as?4<?7 - 4a^q^q^ + 2dbq^ - 4a^bq^ - 4abfiq^ + Sa^bfiq^ - 4a'b/3q^ ) 

A^,, = -^ (a'b -4a'ab + 5aWb - 2aWb + b'- 3ab' + 2a^b' - 2a^abJ3 + 
\-a 

SaWbfi - lOa'a'b/3 + 4a'a'bfi + 2blf - 4ablf + 2a^blf - 2ab/3lf + 2a'bJ3lf - 
2bll + loijll - la'^bll + 2a'bll + 2abl3ll - 6a^bpll + 4a'bpll + bll - 4abll + 
Sa^bll - 2a'bll - 2ab/3ll + 6a^bpll - 4a'bpll - bll + (^^l + 2(Xbl3ll - 2a^b/3ll 



142 



a^q,+6a^ceq^-\3aWq,+\2aWq,-4a^a*q,-b^q,+3odj^q,-2a^b^q,+ 
2b' Pq, - eab'fiq, + 4a'b'fiq, - 2b'/3'q, + Gob'fl'q, - 4a'b'fi'q, + 2l^q, - 
lodlq, + laHlq, - 2aHlq, - llq, + Aodlq, - SaH^q, + 2aHlq, + l^q, - 
30dlq, + 2aHlq, - 2bql + 6C(bql - Aa'bq] + 2b Pq] - Aoeb^q] + 2b' Pq] - 
2ab'pq, - A0d)'P'q, + Aa'b'p'q, - 2bpq,q, + 6oijpq,q, - Aa'bpq.q, + 
a'aq^ - Sa'a'q^ + SaWq^ - 4aWq, - 4b'q, + 1 locb'q^ - 6a'b'q^ + 
Aab'Pq, - SaVPq, - 2b' P\ + 6ab'P\ - Aa'b'P'q, + 2l^q, - 6od^q, + 
Sa'llq, - 2aH^q, + Od^q, - Za'llq, + 2a' lU, - Odlq, + 2a'llq, + 2bq,q, - 
6abq^q^ + Aa'bq.q^, - 2od)pq^q^ + 4a'bj3q,q^ + 2ocbj3q^q^ - Aa'bPq^q^ - 
2bql + Aoi)ql + 2oi)pql - Aa'bpq] + a'q^ - 5a'aq^ + XQa'a'q^ -Wa'a'q^ + 
4a V^7 + 3b' q^ -Wocb'q^ +\Q)a'b'q, - Aa'b'q, - Ab' pq, + Wab'pq^ - 
2a'b'pq^ - Aa'b'pq, + 2b'P'q, - 2(Xb'p'q, - ^a'b'P'q, + Sa'b'p'q, + 
20d^q, - 2a'lUn - 2l'qi + ^od'q^ - 2a'l^q, + l^q^ - 3odlq, + 2a'llq, - ijq^ + 
2a/g ^7 - 2a'llq.j - 2bq^q^ + %odjq^q^ - XQa'bq^q^ + Aa'bq^q^ + 2bpq^q^ - 
6od?Pq^q^ + ea'bpq^q^ - Aa'bpq^q^ - 2aqlq^ + Aa'qlq^ + 2bpq^q^ - 
Aodjpq.q, + Aa'bpq.q^ - 2a'bq^q, + Aa'bq^q^ + Abpq^q, - 1 6od)Pq^q^ + 
1 %a'bpq^q^ - Aa'bpq^q^ - 2q,q^q^ + 6aq^q^q^ - Aa'q^q^q^ + 2bq^ - 6cdyq' + 
Aa'bq' - Abpq', + 1 2abpq' - Sa'bpq^ + 2b' q^ - 6odj'q, + Aa'b'q, - 2b' Pq, - 
2(Xb'pq^ + Sa'b'Pq^ + Aodj'p'q^ - Aa'b'p'q^ - 2bq^q^ + 6ocbq^q^ - Aa'bq^q^ + 
2bpq,q^ - 6abpq,qg + Aa'bpq^q, + 2abq,q, - Aa'bq^q^ - 2cd)Pq^q, + 
Aa'bpq^q^ + 2bq,q^ - Acxbq^q^ - Ibpq^q^ + Aocbpq^q^ - Aa'bpq.q^ ] 

N^ = a^(- a" + %a'a - 25a* a' + 38^ V' - 28fl V + Sa'a' - 2a'b' + 6a'ab' - 
Aa'a'b' -b' + 2ab' -Aa'b'p' +\6a'ab'p' -20a'a'b'P' +Sa'a'b'p' - 
Ab'l^ + Aab'lf - Ab'P'lf + Aab'p'l^ + 40^/3' - 1 Sa'od^ + 30a'a'l^ - 2Aa'aHl + 
%a'aHl+Ab'll - moi}'ll+Aa'b'll+Ab'p'll-\2od)'p'll+%a'b'p'll + 
Al^ll - Aod'l' - 4/' +120/3' - 9a'/' + 20^/3 - 2a' l^ + \2a'odl - 26a' a' ll + 
2Aa'aHl - Sa'aW, - 2b'll + 6ab'll - Aa'b'll - Ab'p'lj + Uab'p'l^ - 



143 



%aWp'll + Aim - ^"^lll + ^^oc^ll - ^ocWe - ll + 4< - 5aX + 2a'C + 
la^l - XOa^odl + Xea'a^ - ^aWl^ + Ib^l + Ab'' fi^ - Aab'fi'l^ - Al^ + 
eodlll - Aa'l^ + 21^1^ - eodUl + AaHlll - 1* + 2< + Aa'bPq, - 20a'od?/3q, + 
32aWb^,-l6aWbfiq, -Ab'fiq,+S(xb'j3q,+Aab/3l',q, -Sa'bfil^q, + 
Abpllq^ - ncdj/Mlq, + M^bjMlq, - Abfil^q, + Sobfy, + Ab^ql - 80*^^4' - 
4/3^^4' + Sod^ql - Aa'bfiq, + 20a^abPq, - 32aWb/3q, + I6a'a'bfiq, - Ab'j3q, - 
Sb'fi'qs + Sab'fi'q, + Sbfil^q^ - I2abj3l^q, + Sa^b/3l^q, - Abpllq, + 
nocbpllq, -Sa'bfillq,+Abfil^q,-^abj3l^q,+Sb^P'q,q,-l6ab^P^q,q, - 
Ab^fi^ql + Sab^fi^l + Aa'bq^ - 20a^abq^ + 32aWbq^ - \6a^a^bq^ + Ab^q^ - 
Sab^q^ + Sb^P^qf, - \6ab'fi^q^ - Sbl^q^ + 20abl^q, - Sa^bl^q^ + Abl^q^ - 
nabl^q, + Sa'bllq, - Abllq, + %CXbllq, - Sb'j3q,q, + l6(Xb'j3q,q, + Sb'/3q,q, - 
\6ccb^pq,q,+Ab^P\l -Sob'fiY, -Al^ql +Sod^ql -Aa^bq, +2Aa^abq, - 
52aWbq.j + ASa^a^bqy -l6aWbq^ -Ab^q.+Uob^q^ -Sa^b^q^ +Sa^b/3q^ - 
36a^abJ3qT+60a^a^bfiq, -ASaWb/^q^ +I6a^a'b/3q, +Sb^/3q^ -20ab^J3q^ + 
Sa^b'fiq, -Sb'fi^q^ +2Aab'fi^qT -lea'b'fi^q^ + %b' P\, - 2Aocb^ p\, + 
lea^b'fi'q^+Sbfilfq^ -Sotbfil^q, +Sbl^qT -2^abl^qT +2Sa^bl^q^ -Sa'bl^q^ - 
\6bpllq,+A%oi}pllq, -36a^b/3l^q, +Sa^b/3l^q., -Abl^q^ +I6abllq^ - 
20arX^7 +^oc^bllq, +%b/3l^qT -2Socb/3llqy +2^^bfy^ -Sa^b/M^q^ + 
Abllq, -nodjllq, +2,a^bllq^ -%bpilq,+\2cd)pllq^ -%a^bpllq,+%b^pq^q^ - 
2A(Xb^j3q,qj+l6a^b^/3q,q, +Sab^j3^q,q,-I6a^b^fi^q,qj -Sbfiqlq^ + 
Itabpqlq^ -W^ pq.q^ +2A0i}^ pq^q, -\6a^b^ pq.q^ +\6b^ P^q,q^ - 
2A(xb^ P^q^q^ +\6aW p^q^q^ -Aa^q^q^ +20a^aq^q^ -32a^a^q^qj + 
16aV^6<?7 + Ab^q^qT -IGab^q^qy +\6aWq^qT -I6b^ Pq^qj + AOod?^ Pq^q^ - 
l6a'b'Pq,q, -Sb'P'q.q, +l6ab'P\q, +Sl^q,q, -20al^q,q, +SaY,q,q, - 
Allq^q^ + 1 2cdlq^q^ - SaH^q^q^ + Al^q.q, - Sod^q^q^ + SbPq.q^q^ - 
1 6abPq,q,q^ - Sbpq.q^q^ + 1 6abpq,q^q, + Wqlq, - 1 eoijqlq^ - SbPqjq^ + 
\6odjPqlq-, -Aa^aq^+Ua^a^q^ -SaWqT -Ab^q^ +\6ocb^qT -20a^b^q^ + 
Sa'b^q^ + l6b'Pq^ - 56cxb^pq^ + 56aWpq'^ - l6a'b^Pq^ - \2b''p\] + 



144 



40ab'fi'q^-36aVp'q^ +Sa'b'/3'q^ -4l^q^ +4odfq', +4cd^q', Sb^.q^ 4 
l6(Xbfiq,q^ + AqW^ - %aq\q] - ^do^q^q] - Wq^] + 240*^,^7^ -\6a'bq,q^, + 
\6bJ3q,q^ - 40abfiq,q^ + I6a'bpq(,q', - 4a^bq, + lOa^obq, - 32a'a'bq, + 
\6aWbq^ - 4b^q^ + Socb^q^ + 4a^bl3q^ - 20a^0(bpq^ + 32aWbfiq^ - 
\6aWb/3q,+4b'^, -Sb'fi\+\6c(b'fi'q,+Sb'fi'q, -Sab'j3'q,+SbP,q, - 
lOocbl^q, + Sa^bl^q, - Sbfil^q, + 1 lodyfil^q, - Sa'b/3l^q, - 4bl^q, + 1 2ocbl^q, - 
^a^bllq^ + 4bpllq^ -\2ab/M^,q, + Sa^bpl^q, + 4bllq^ - Sod^l^q, - 4bj3l^q, + 
SdyPl^q,+Sb'/3q,q,-\60(b'/3q,q,-ib'fi\q,+l6ab'l3'q,q, Sb'Pq.q, + 
\6c(b'/3q,q,+Sb'j3'q,q,-l6ab'fi\,q,+Sb\q,-\6ab^q,q, -Sb'j3q,q, + 
\6c(b'/3q^qs -Wq^q^+24oi}^q,q^-\6aWq,q^+24b''l5q,q^-64(Xb''pq,q^ + 
32a^b^ I3q,q,-\6b^ P\-,q^+24oi)^ P^q,q,-\6aW P^.q^ -Sbq,q,q^ + 
\6(Xbq,q,q, + Sbj3q,q^q, -l6abPq,q,q, + Socbfiq^q, - 4b\l + %ab'ql + %b^ Pq] 
\6ab'fiq', -4b'P'ql +Sody'/3Vs) 



N.. = 



4a\\-2ay 



;» h 



(i-ay 

^ 4a\\-2af /_ ^^,^ + ga V - 6aW + 2a'a' + 2b' /3' - 4cd,'/3' + 2a'b'/3' - 

If + 2od^ - a'lf - ll + 2odl - aHl - 2bpq, + \Qoi,pq, - ^a'b^, + a\l - 6abPq, + 
6a'bJ3q, - 2aq,q, + q] - 4abJ3q, + 4a'bj3q, - 2094^6 + 2a'?496 + ^Qs^e ' 4«?5^6 + 
ql - 2aq\ + a'ql - 2bpq^ + Sc^yS^g - ba'bfiq^ - 2q,q^ + 2aqSi ) 

1-a 
13aV94 -12aV^4 +4a V^4 + 2b^fi'q,-6ab'P'q,+4aWfi'q,-l^q, + 

2od^q, - a'lfq, - llq, + 4(Xllq, - 3a'l^q, - 2bPql + SO(bfiql - Sa'b/3ql + 

2a'aq,-SaWq,+l0aWq, -4a'a'q,-2b'fi'q,+6ab'fi'q, -4aWj3\ + 



145 



^50 



2l^q, - Acd^q, + la'^l^s " '^U^ + ^OC^lq, + Abpq.q, - 1 %Cd}pq,q, + 
\Sa^bj3q,q, + 6oi?yQ^5^ - Sa^b/3q^^ + a^aq, - Sa^a^q^ + ^a^a'q^ - 4aWq^ - 
2b'fi\ + 6ab'j3% - 4aVp'q, - Odfq, + a'lfq, + lljq, - 5od^q, + Sa'l^q, - 
lcdjl3q,q^ + Aa^bpq.q^ - la^bpq.q^ + loq^qsq^ " 4«'^495^6 " 2^596 + ^mU^ + 
Icdjpql - Aa^bpql - 2q,ql + 6aq,ql - Aa^.q] + a'q^ - Sa'oq^ + \OaWq, - 
lOflV^g +Aa^a'q, + 2b^/3\-6ab'P^q,+Aa^b^fi\, -lfq,+3od^q, - 
2aHfq, - llq^ + 30dlq^ - 2aH^q, - Abfiq.q, + 1 6abPq,q, - \Aa^bfiq,q, - 
2aqlq^ + Aa^qlq^ - ^abfiq^q^ + Sa^bfiq^q^ + 2q^q^q^ - Aoq.q^q^ + Ab/5q^q^ - 
\Aabfiqf,q^ + XAa^bpq^q^ + 2q,q^q^ - 6aq^q,q^ + Aa^q^q^q^ - 2bfiql + Sobfiq^ - 
%a'bPql) 

= -a^(a'-\Oa'a + AlaW -88aV +104aV -64aV +16aV +4aVy9' - 
2Aa'ab'/3'+52aWb'j3' -ASa'a'b'j3' +l6aWb'fi' -2a'l^ + l2a'od^ - 
26aWlf + 2Aa'a'lf - SaWl^ + /,' - 2od^ + aH^ - 2aHl + 12fl'< - 
26aWll + 2Aa'aHl - Sa'a'l', - 2l^ll + Aod^ll - 2aH^ll + ll - 2al^ + a^ - 
Aa^bjSq, + 2Sa^0dj/3q, - 12aWbfiq, + ^OaWb^q, - 32aWbpq, + Ab/3lfq, - 
UocbplU, + Sa'bfil^q, - Abpljq, + \2od?pllq, - Sa^bJJl^q, + Al^qj -I6al^ql + 
\6a'liql -Aa'ab/3q, + 20aWbPq, - 32aWbfiq, + l6aWb/3q, + Aabfil^q, - 
Aa^bfilfq, -Aabpllq^ +Aa^bfil^q^+Aa^q^q,-2Sa^0iq,q^ +12aWq,qs - 
SOaWq^q, + 32a^a'q^q^ + Sb^fi^^q^ - 2Aoi}^ p^^q^ + I6a^b^fi^q,q, - 
Alfq,q,+I2al^q,q, -Sa^lfq^q^ -Al^q,q, + 20odiq,q,-2Aa'l^q,q, Sbfiqlq, + 
32oi)pqlq,-32a^bpq]q, +Aa^aql-20aWql+32a^a^q^^ -IGaWq^ - 
Ab'j3'ql+\6ab'/3'q',-l2a'b'/3'ql+Alfq',-l2od^ql +Sa'l^qj -Aodlq] + 
%aHlql + 9,bpq,ql - A0abJ3q,qj + ASa'bjSq.qj + Sab/3ql - 1 6a^b/iql - 
Ab'fi'ql + leab'fiVe -l6aV/3'ql + Aljql - l6od',ql + \6aHlql - Aqlq] + 
\eaqlql-\6a^qlql -Aa^b^^ + 2Aa^abfiq^ -52a^a^bPq, +ASaWb/3q, - 
I6a^a*bfiq^ - Abfilfq^ + Socbfilfq^ - Aa^bj3lfq^ + Abfil^q, - ^obpllq^ + 
Aa'bfil^q, - Sb'fi'q.q, + 2Ac(b'fi\q, -\6aWj3\q, + Sbj3qlq, - 



146 



32cd)Pqlq,+32a-bJ3qlq,-Sab'fi'q,q,+Sa'b'fi^q,q,-Sbfiq,q,q, + 
40(XbPq,q,q,-4Sa'bj3q,q,q, -Sabfiq^q, +\6a'bPqlq, +4a^q,q, -ISa^oq.q, 4 
12aWq,q,-S0aWq,q,+32a^a'q^q,+^b^fi\q,-32ab^fi^q,q, + 
32aWfi'q,q,-4lfq,q,+l2odfq,q,-SaH^q,q, -4l',q,q, +2Qodiq,q, - 
24aHlq^q, - 8bfiq,q,q, + 32abfiq,q,q, - 32a'bJ3q,q,q, - S(XbPq,q,q, + 
\6a^b/3q,qf,q, + Sq.q.q^q^ - 32aq,q,q(^q^ + 32a\^q^qf,q^ + ^b^qlq^ - 
32ab/3qlq, + 32a'bJ3qlq, + 4a^aql - 20aWql + 32aWql -\6a'a*q^ + 
Sab'PV,-\2a'b'0Vs +4iy,-l2ody,+Sa'lfq', -4odlql +Sa'l',q', + 
WPq.ql - 32oi}Pq,ql + 32a'bfiq,ql - 4qlql + \eaqlql - \6a'qlql + SabJ3q,q~ 
I6a'bj3q,ql -SbJ3q,ql +40abflq,ql -4Sa'bl3q,ql -Sab/3ql +\6a'bfiql) 



N.. = 



16a' 



A^g3 = ^-^-^ib-2ab + a^b-2ab/3 + 2a^bJ3 + a^qs+oaiT-a^qT+oai^-a^q^) 

\i CX) 

N,,= ^{-2a' +Sa'a-lOaW +4aW -b'+2od?' -«V - 4b' J3' + Sod?' J3' - 

4aV/3' + 2ll - 4cdl + 2a'll + 2ll - 4odl + la'll - 6bq, +]2abq, - 6a'bq, + 
Sotbfiq^ - Sa'bfiq^ - a'q] + 2oi)q^ - 2a'bq^ + 4bpq^ - Sabfiq^ + 4a'bpq^ - 60(q^q^ + 
6a^9597 - <?7 + 2aq] - a'q] - 2oi)q^ + 2a'bq^ + 4bpq^ - Sabpq^ + 4a'b/3q^ - 
2aq^q^ + 2a'q^q^ - 2q^q^ + 4aq^q^ - 2a'q^q^ - q] + 20^8 - a'ql ) 



N., =- 



\-a 



(- 2a'b + 6a'od? - 6a'a'b + 2a'a'b -b'+(xb'+ 4a'cd}fi - Sa'a'bfi + 

Aa'a^bp + bl] - cebl' - bll + cdyll + 2a' q^ - 6a'oeq^ + 6a'a'q^ - 2aWq^ + 
3b' q, - 3ab'q, + 4b'P'q, - 4ceb'fi'q, - 3l^q, + 3al^q, - l^q, + Od^q, + 2bq', - 
2abq' - 2abfiq^ - 2a'aq^ + 4a'a'q^ - 2a'a^q^ - ab'q^ + 2b' pq, - 2ab'/Jq^ + 
Odlq, - Odlq, - ebpq^q^ + 6cxb/3q,q^ + 2aqlq, + 2q^q'^ - 2aq^q^ - 2a'aq^ + 
4a'a'q^ - 2a'a'q^ + CXb'q^ - 2b' pq^ + 2od?'pq, - od^q, + Od^q, - 2bfiq,q, + 
2(Xbl3q^q^-2bq^q^+2od)q^q^+2q^q^q^-2aq,q^q^+2bql-2cd)ql) 



147 



N^ = -4a' + \6a'a - 24aW + 16a V - 4a V - 4a '^' + Sa'cxb' - 4aWb' - b' - 
Sa'b'fi'+l6a'ab'P' -SaWb'fi'+4a'l', -Sa'od^ +4aWl^ + 2b'l', -it + 
4aHl - %a^0dl + 4aWll - Ib'll + lllll - it + 4b' q] + 4b'p'ql - 4l',q, + 
Sa'bPq, -I6a'ab/3q, +&aWbPq, +4b'/3q, -4b^lq, +4bpilq, Sbfiq^q, - 
4iy, + 4qWi + ^a'bPq, - I6a'abj3q, + SaWbj^q, - 4b'fiq, + 4bfil^q, - 
4bfil^q^ -8a'97^8 +16a'Q97^8 -%a'a'q^q^ -4b'q^q^+4l^q^q^+4l^q^q^ + 



2^2 



4bV,-^lks 



*■-►• J 



A ,:; 



■- * . f 



LIST OF REFERENCES 

1. Ball, R. S., A Treatise on the Theory of Screws, Cambridge University Press, 
London, 1900. 

2. Buttolo, P., and Hannaford, B., "Advantages of Actuation Redundancy for the 
Design of Haptic Displays," Proceedings of ASME Fourth Annual Symposium on 
Haptic Interfaces for Virtual Environment and Teleoperator Systems, Vol. DSC 
57-2, pp. 623-630, San Francisco, November 1995. 

3. Buttolo, P., and Hannaford, B., "Pen-Based Force Display for Precision 
Manipulation in Virtual Environments," Proceedings of IEEE Virtual Reality Annual 
International Symposium, pp. 217-225, Raleigh, North Carolina, March 1995. 

4. Cox, D. J., "The Dynamic Modeling and Command Signal Formulation for Parallel 
Multi-Parameter Robotic Devices," Master Thesis, Mechanical Engineering, 
University of Florida, 1981. 

5. Crane, C, and Duffy, J., Kinematic Analysis of Robot Manipulators, Cambridge 
University Press, 1998. • . 

6. Dasgupta, B., and Mruthyunjaya, T. S., "Force Redundancy in Parallel Manipulators: 
Theoretical and Practical Issues," Mechanism and Machine Theory, Vol. 33, No. 6, 

pp. 727-742, 1998. - , . . ,- 

7. Dasgupta, B., and Mruthyunjaya, T. S., "The Stewart Platform Manipulator: A 
Review," Mechanism and Machine Theory, Vol. 35, No. 1, pp. 15-40, 2000. 

8. Duffy, J., Statics and Kinematics with Applications to Robotics, Cambridge 
University Press, London, 1996. 

9. Duffy, J., Rooney, J., Knight, B., and Crane, C, "A Review of a Family of Self- 
Deploying Tensegrity structures with Elastic Ties," The Shock and Vibration Digest, 
Vol. 32, No. 2, pp. 100-106, 2000. 

10. Fichter, E. F., "A Stewart Platform-Based Manipulator: General Theory and 
Practical Construction," The International Journal of Robotics Research, Vol. 5, pp. 
157-182, 1986. 



148 



149 



11. Gosselin, C, and Angeles, J., "The Optimum Kinematic Design of a Planar Three- 
Degree-of-Freedom Parallel Manipulator," ASME Journal of Mechanisms, 
Transmissions, and Automation in Design, Vol. 1 10, No. 3, pp. 35-41, 1988. 

12. Gosselin, C, and Angeles, J., "The Optimum Kinematic Design of a Spherical 
Three-Degree-of-Freedom Parallel Manipulator," ASME Journal of Mechanisms, 
Transmissions, and Automation in Design, Vol. Ill, No. 6, pp. 202-207, 1989. 

13. Hunt, K. H., Kinematic Geometry of Mechanisms, Oxford University Press, London, 
1978. 

14. Hunt, K. H., and McAree, P. R., "The Octahedral Manipulator: Geometry and 
Mobility," The International Journal of Robotics Research, Vol. 17, No. 8, pp. 868- 
885, 1998. 

15. Innocenti, C, "Closed-Form Determination of the Location of a Rigid Body by 
Seven In-Parallel Linear Transducers," Proceedings of the ASME 24th Biennial 
Mechanisms Conference, Irvine, Ca., 1996. 

16. Knight, B., Zhang, Y., Duffy, J., and Crane, C, "On the Line Geometry of a Class of 
Tensegrity Structures," Proceedings of Sir Robert Stawell Ball 2000 Symposium, 
University of Cambridge, UK, July 2000. 

17. Kock, S., and Schumacher, W., "A Parallel X-y Manipulator with Actuation 
Redundancy for High Speed and Active Stiffness Applications," IEEE Int. Conf. on 
Robotics and Automation, pp. 2295-2300, Louvain, 1998. 

18. Kokkinis, T., and Millies, P., "A Parallel Robot-Arm Regional Structure with 
Actuational Redundancy," Mechanism and Machine Theory, Vol. 26, No. 6, pp. 
629-641, 1991. 

19. Kurtz, R., and Hayward, V., "Multiple-Goal Kinematic Optimization of a Parallel 
Spherical Mechanism with Actuator Redundancy," IEEE Transactions on Robotics 
and Automation, Vol. 8, No. 5, pp. 644-651, October 1992. 

20. Lee, J., and Duffy, J., "The Optimum Quality Index for Some Spatial In-Parallel 
Devices," 1999 Florida Conference on Recent Advances in Robotics, Gainesville, 
FL, April 1999. 

21. Lee, J., Duffy, J., and Hunt, K. H., "A Practical Quality Index Based on the 
Octahedral Manipulator," The International Journal of Robotics Research, Vol. 17, 
No. 10, pp. 1081-1090, 1998. 

22. Lee, J., Duffy, J., and Keler, M., "The Optimum Quality Index for the Stability of 
In-Parallel Planar Platform Devices," Proceedings of the ASME 24th Biennial 
Mechanisms Conference, 96-DETC/MECH-1135, Irvine, Ca., 1996. 



150 



23. Leguay-Durand, S., and Reboulet, C, "Optimal Design of a Redundant Spherical 
Parallel Manipulator," Robotica, Vol. 15, No. 4, pp. 399-405, 1997. 

24. Maeda, K., Tadokoro, S., Takamori, T., Hiller, M., and Verhoeven, R., "On Design 
of a Redundant Wire-Driven Parallel Robot WARP Manipulator," IEEE Int. Conf. 
on Robotics and Automation, pp. 895-900, Detroit, May 1999 

25. Merlet, J. P., "Redundant Parallel Manipulators," Laboratory Robotics and 
Automation, Vol. 8, No. 1, pp. 17-24, 1996. 

26. Merlet, J. P., "Singular Configurations of Parallel Manipulators and Grassmann 
Geometry," The International Journal of Robotics Research, Vol. 8, No. 5, pp. 45- 
56, 1989. 

27. Nakamura, Y., and Ghodoussi, M., "Dynamics Computation of Closed-Link Robot 
Mechanisms with Nonredundant and Redundant Actuators," IEEE Transactions on 
Robotics and Automation, Vol. 5, No. 3, pp. 294-302, June 1989. 

28. Notash, L., and Podhorodeski, R. P., "Forward Displacement Analysis and 
Uncertainty Configurations of Parallel Manipulators with a Redundant Branch," 
Journal of Robotic Systems, Vol. 13, No. 9, pp. 587-601, 1996. 

29. O'Brien, J. F., and Wen, J. T., "Redundant Actuation for Improving Kinematic 
Manipulability," IEEE Int. Conf. on Robotics and Automation, pp. 1520-1525, 
Detroit, May 1999. 

30. Pemg, M., and Hsiao, L., "Inverse Kinematic Solutions for a Fully Parallel Robot 
with Singularity Robustness," The International Journal of Robotics Research, Vol. 
18, No. 6, pp. 575-583, June 1999. 

31. Raghavan, M., "The Stewart Platform of General Geometry has 40 Configurations," 
ASME Journal of Mechanical Design, Vol. 1 15, No. 2, pp. 277-282. 

32. Selfridge, R. G., "About some Tensegrity Structures," submitted to the Journal of 
Space Structure, 2000. 

33. Shin, J., and Lee, J., "Fault Detection and Robust Fault Recovery Control for Robot 
Manipulators with Actuator Failures," IEEE Int. Conf. on Robotics and Automation, 
pp. 861-866, Detroit, May 1999. 

34. Stewart, D., "A Platform with Six Degrees of Freedom," Proceedings of the 
Institution of Mechanical Engineers, Vol. 180, No. 1, pp. 371-386, 1965. 

35. White, N., and Whiteley, W., "The Algebraic Geometry of Stresses in Frameworks," 
S.I.A.M. Journal of Algebraic and Discrete Methods, Vol. 4, No. 4, pp. 481-511, 
1983. 



151 



36. Zanganeh, K. E., and Angeles, J., "Kinematic Isotropy and the Optimum Design of 
Parallel Manipulators," The International Journal of Robotics Research, Vol. 16, 
No. 2, pp. 185-197, 1997. 

37. Zhang, Y., Crane, C, and Duffy, J., "Determination of the Unique Orientation of 
Two Bodies Connected by a Ball-and-Socket Joint From Four Measured 
Displacements," Journal of Robotic Systems, Vol. 15, No. 5, pp. 299-308, 1998. 






-S :; { J 



BIOGRAPHICAL SKETCH 

Yu Zhang was bom on September 16, 1969, in Beijing, People's Republic of 
China. He received the Bachelor of Engineering degree in Mechanical Engineering from 
Dalian University of Technology in August 1992 and the Master of Engineering degree 
in Mechanical and Electronic Engineering from Beijing University of Posts and 
Telecommunications in April 1995. He began his Ph.D. study in Mechanical Engineering 
at the University of Florida in August 1996. 



152 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




Josegji Duffy, Ch 
Graduate Research Pibfessor of 
Mechanical Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




r 



Carl D. Crane m 

Professor of Mechanical Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




u 



jLJg/VJ^ 



Gloria Jt,JA^iens 

Associate Professor of Mechanical 
Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 



L\A^ 




Ali A. Seirig 

Ebaugh Professor of Mechanical 
Engineering 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 




RalgltJSelfridj 
Professor of Compdter and Information 
Science and Engineering 

This dissertation was submitted to the Graduate Faculty of the College of 
Engineering and to the Graduate School and was accepted as partial fulfillment of the 
requirements for the degree of Doctor of Philosophy. 



December 2000 



r 



0- 



M. Jack Ohanian 

Dean, College of Engineering 



Winfred M. Phillips 
Dean, Graduate School 



1 i 



,^- .■ 




20 00 



UNIVERSITY OF FLORIDA 

3 1262 08555 1728